Chapter 4 Resource Masters - OpenStudy

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Geometry Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term ... This is a list of key theorems and postulates you will learn in Chapter 4. As you study the .... Study Guide and Intervention (continued). Classifying ...
Geometry Chapter 4 Resource Masters

NAME ______________________________________________ DATE

____________ PERIOD _____

Reading to Learn Mathematics

4

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter. Vocabulary Term

Found on Page

Definition/Description/Example

acute triangle

base angles

      

congruence transformation kuhn·GROO·uhns

congruent triangles

coordinate proof

corollary

equiangular triangle

equilateral triangle

exterior angle

(continued on the next page) ©

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Vocabulary Builder

Vocabulary Builder

NAME ______________________________________________ DATE

____________ PERIOD _____

Reading to Learn Mathematics

4

Vocabulary Builder Vocabulary Term

(continued)

Found on Page

Definition/Description/Example

flow proof

included angle

included side

isosceles triangle

obtuse triangle

remote interior angles

right triangle

    

scalene triangle SKAY·leen

vertex angle

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NAME ______________________________________________ DATE

4

____________ PERIOD _____

Learning to Read Mathematics

This is a list of key theorems and postulates you will learn in Chapter 4. As you study the chapter, write each theorem or postulate in your own words. Include illustrations as appropriate. Remember to include the page number where you found the theorem or postulate. Add this page to your Geometry Study Notebook so you can review the theorems and postulates at the end of the chapter. Theorem or Postulate

Found on Page

Description/Illustration/Abbreviation

Theorem 4.1 Angle Sum Theorem

Theorem 4.2 Third Angle Theorem

Theorem 4.3 Exterior Angle Theorem

Theorem 4.4

Theorem 4.5 Angle-Angle-Side Congruence (AAS)

Theorem 4.6 Leg-Leg Congruence (LL)

Theorem 4.7 Hypotenuse-Angle Congruence (HA)

(continued on the next page) ©

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Proof Builder

Proof Builder

NAME ______________________________________________ DATE

4

____________ PERIOD _____

Learning to Read Mathematics Proof Builder Theorem or Postulate

(continued) Found on Page

Description/Illustration/Abbreviation

Theorem 4.8 Leg-Angle Congruence (LA)

Theorem 4.9 Isosceles Triangle Theorem

Theorem 4.10

Postulate 4.1 Side-Side-Side Congruence (SSS)

Postulate 4.2 Side-Angle-Side Congruence (SAS)

Postulate 4.3 Angle-Side-Angle Congruence (ASA)

Postulate 3.4 Hypotenuse-Leg Congruence (HL)

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NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-1

Classifying Triangles Classify Triangles by Angles

of its angles.

One way to classify a triangle is by the measures

• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle. • If one of the angles of a triangle is a right angle, then the triangle is a right triangle. • If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.

Example a.

Lesson 4-1

• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.

Classify each triangle.

A

B

60!

C

All three angles are congruent, so all three angles have measure 60°. The triangle is an equiangular triangle. b.

E 120! 35!

D

25!

F

The triangle has one angle that is obtuse. It is an obtuse triangle. c.

G 90!

H

60!

30!

J

The triangle has one right angle. It is a right triangle.

Exercises Classify each triangle as acute, equiangular, obtuse, or right. 1. K

2. N

30!

67!

L

4.

23!

3.

Q

65! 65!

60!

P

M

R

5. W

60!

60!

V

Glencoe/McGraw-Hill

X

S

6.

B 60!

45!

50!

©

O

120!

90!

T

U

30!

90!

45!

Y

183

F

28!

92!

D

Glencoe Geometry

NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-1

(continued)

Classifying Triangles Classify Triangles by Sides You can classify a triangle by the measures of its sides. Equal numbers of hash marks indicate congruent sides. • If all three sides of a triangle are congruent, then the triangle is an equilateral triangle. • If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle. • If no two sides of a triangle are congruent, then the triangle is a scalene triangle.

Example a.

Classify each triangle. b.

H

c.

N

T 23

L

R

J

Two sides are congruent. The triangle is an isosceles triangle.

X

P

All three sides are congruent. The triangle is an equilateral triangle.

12

V

15

The triangle has no pair of congruent sides. It is a scalene triangle.

Exercises Classify each triangle as equilateral, isosceles, or scalene. 1.

A 2

G

4.

!" 3

2.

1

C

18

K

5. B

S

3.

G

18

W

17

12

18

Q

I

A 32x

O

19

6. D

32x

8x

C

M

x x

E x

F

U

7. Find the measure of each side of equilateral !RST with RS ! 2x " 2, ST ! 3x, and TR ! 5x # 4. 8. Find the measure of each side of isosceles !ABC with AB ! BC if AB ! 4y, BC ! 3y " 2, and AC ! 3y.

9. Find the measure of each side of !ABC with vertices A(#1, 5), B(6, 1), and C(2, #6). Classify the triangle.

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NAME ______________________________________________ DATE

4-1

____________ PERIOD _____

Skills Practice Classifying Triangles

1.

2.

3.

4.

5.

6.

Identify the indicated type of triangles. 7. right

B

A

8. isosceles E

9. scalene

Lesson 4-1

Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

D

C

10. obtuse

ALGEBRA Find x and the measure of each side of the triangle. 11. !ABC is equilateral with AB! 3x # 2, BC ! 2x " 4, and CA ! x " 10.

12. !DEF is isosceles, "D is the vertex angle, DE ! x " 7, DF ! 3x # 1, and EF ! 2x " 5.

Find the measures of the sides of !RST and classify each triangle by its sides. 13. R(0, 2), S(2, 5), T(4, 2)

14. R(1, 3), S(4, 7), T(5, 4)

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NAME ______________________________________________ DATE

4-1

____________ PERIOD _____

Practice Classifying Triangles

Use a protractor to classify each triangle as acute, equiangular, obtuse, or right. 1.

2.

3.

Identify the indicated type of triangles if !B A !"A !D !"B !D !"D !C !, B !E !"E !D !, A !B !⊥B !C !, and E !D !⊥D !C !. 4. right

B E

5. obtuse A

6. scalene

D

C

7. isosceles

ALGEBRA Find x and the measure of each side of the triangle. 8. !FGH is equilateral with FG ! x " 5, GH ! 3x # 9, and FH ! 2x # 2.

9. !LMN is isosceles, "L is the vertex angle, LM ! 3x # 2, LN ! 2x " 1, and MN ! 5x # 2.

Find the measures of the sides of !KPL and classify each triangle by its sides. 10. K(#3, 2) P(2, 1), L(#2, #3)

11. K(5, #3), P(3, 4), L(#1, 1)

12. K(#2, #6), P(#4, 0), L(3, #1)

13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there?

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NAME ______________________________________________ DATE

____________ PERIOD _____

Reading to Learn Mathematics

4-1

Classifying Triangles Pre-Activity

Why are triangles important in construction? Read the introduction to Lesson 4-1 at the top of page 178 in your textbook. • Why are triangles used for braces in construction rather than other shapes?

Reading the Lesson 1. Supply the correct numbers to complete each sentence. a. In an obtuse triangle, there are

acute angle(s),

right angle(s), and

obtuse angle(s). b. In an acute triangle, there are

acute angle(s),

right angle(s), and

obtuse angle(s). c. In a right triangle, there are

acute angle(s),

right angle(s), and

obtuse angle(s). 2. Determine whether each statement is always, sometimes, or never true. a. A right triangle is scalene. b. An obtuse triangle is isosceles. c. An equilateral triangle is a right triangle. d. An equilateral triangle is isosceles. e. An acute triangle is isosceles. f. A scalene triangle is obtuse. 3. Describe each triangle by as many of the following words as apply: acute, obtuse, right, scalene, isosceles, or equilateral. a.

b.

70! 80!

30!

c. 135!

4

3 5

Helping You Remember 4. A good way to remember a new mathematical term is to relate it to a nonmathematical definition of the same word. How is the use of the word acute, when used to describe acute pain, related to the use of the word acute when used to describe an acute angle or an acute triangle?

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Lesson 4-1

• Why do you think that isosceles triangles are used more often than scalene triangles in construction?

NAME ______________________________________________ DATE

4-1

____________ PERIOD _____

Enrichment

Reading Mathematics When you read geometry, you may need to draw a diagram to make the text easier to understand.

Example

Consider three points, A, B, and C on a coordinate grid. The y-coordinates of A and B are the same. The x-coordinate of B is greater than the x-coordinate of A. Both coordinates of C are greater than the corresponding coordinates of B. Is triangle ABC acute, right, or obtuse? To answer this question, first draw a sample triangle that fits the description. Side AB must be a horizontal segment because the y-coordinates are the same. Point C must be located to the right and up from point B. From the diagram you can see that triangle ABC must be obtuse.

y

Q

A

B O

x

Answer each question. Draw a simple triangle on the grid above to help you.

©

1. Consider three points, R, S, and T on a coordinate grid. The x-coordinates of R and S are the same. The y-coordinate of T is between the y-coordinates of R and S. The x-coordinate of T is less than the x-coordinate of R. Is angle R of triangle RST acute, right, or obtuse?

2. Consider three noncollinear points, J, K, and L on a coordinate grid. The y-coordinates of J and K are the same. The x-coordinates of K and L are the same. Is triangle JKL acute, right, or obtuse?

3. Consider three noncollinear points, D, E, and F on a coordinate grid. The x-coordinates of D and E are opposites. The y-coordinates of D and E are the same. The x-coordinate of F is 0. What kind of triangle must !DEF be: scalene, isosceles, or equilateral?

4. Consider three points, G, H, and I on a coordinate grid. Points G and H are on the positive y-axis, and the y-coordinate of G is twice the y-coordinate of H. Point I is on the positive x-axis, and the x-coordinate of I is greater than the y-coordinate of G. Is triangle GHI scalene, isosceles, or equilateral?

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NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-2

Angles of Triangles Angle Sum Theorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found. The sum of the measures of the angles of a triangle is 180. In the figure at the right, m"A " m"B " m"C ! 180.

B A

Example 1

Example 2

Find the missing angle measures.

Find m"T. S

B

35! 25!

R

C

90!

T

m"R " m"S " m"T ! 180 25 " 35 " m"T ! 180 60 " m"T ! 180 m"T ! 120

Angle Sum Theorem Substitution Add. Subtract 60 from each side.

A

58!

C

1

2

108!

D

3

E

m"1 " m"A " m"B m"1 " 58 " 90 m"1 " 148 m"1

! ! ! !

180 180 180 32

m"2 ! 32 m"3 " m"2 " m"E m"3 " 32 " 108 m"3 " 140 m"3

! ! ! !

180 180 180 40

Angle Sum Theorem Substitution Add. Subtract 148 from each side. Vertical angles are congruent. Angle Sum Theorem Substitution Add. Subtract 140 from each side.

Exercises Find the measure of each numbered angle. 1.

62!

3. V

60!

W U

5.

90!

1

P

2.

M

1

4. M 1 30!

P

6. A

R 30!

Glencoe/McGraw-Hill

R

3

66!

1

T

2

T 60! W

30!

Q

N

1 2

©

S

58!

Q

20! 152!

189

50!

O

G

S

2

N

1

D

Glencoe Geometry

Lesson 4-2

Angle Sum Theorem

NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-2

(continued)

Angles of Triangles Exterior Angle Theorem

At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, "B and "A are the remote interior angles for exterior "DCB. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m"1 ! m"A " m"B

Exterior Angle Theorem

B 1

C

D

Example 1

Example 2

Find m"1.

Find x.

P

S

78! Q

80!

R

A

1

60!

T

m"1 ! m"R " m"S ! 60 " 80 ! 140

Exterior Angle Theorem Substitution Add.

55!

x!

S

R

m"PQS ! m"R " m"S 78 ! 55 " x 23 ! x

Exterior Angle Theorem Substitution Subtract 55 from each side.

Exercises Find the measure of each numbered angle. 1.

2.

X

A 35!

50!

Y

1

65!

Z

3.

N 1 3

Q

O

M

60! 2

2 1

25!

B

W

4.

R 80!

C

V 1

60!

P

S 3

2 35!

U

D

36!

T

Find x. 5.

6. E

A 95!

B

©

2x !

Glencoe/McGraw-Hill

x! 145!

C

D

H

190

58!

G

x!

F

Glencoe Geometry

NAME ______________________________________________ DATE

4-2

____________ PERIOD _____

Skills Practice Angles of Triangles

Find the missing angle measures. 1.

80!

S TIGER

2.

146!

73!

Find the measure of each angle.

85!

55!

1

3. m"1

2

40!

3

4. m"2

Find the measure of each angle.

3

6. m"1

1

2

55!

150!

70!

7. m"2 8. m"3

Find the measure of each angle. 9. m"1

40!

80! 1

60!

4 105!

2

10. m"2

5

3

11. m"3 12. m"4 13. m"5

Find the measure of each angle.

B

14. m"1

1

15. m"2

©

Glencoe/McGraw-Hill

A

191

2

D

63! C

Glencoe Geometry

Lesson 4-2

5. m"3

NAME ______________________________________________ DATE

4-2

____________ PERIOD _____

Practice Angles of Triangles

Find the missing angle measures. 1.

2.

72! ?

40!

55!

Find the measure of each angle.

3

58!

3. m"1

1

2 35!

4. m"2

39!

5. m"3

Find the measure of each angle.

5 2

6. m"1

3

1

7. m"4

70!

36! 68!

118! 6

4 65! 82!

8. m"3 9. m"2 10. m"5 11. m"6

Find the measure of each angle if "BAD and "BDC are right angles and m"ABC " 84.

B

12. m"1 A

1

64! C 2

D

13. m"2 14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridge construction. Use the diagram to find m"1.

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Glencoe/McGraw-Hill

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1

145!

Glencoe Geometry

NAME ______________________________________________ DATE

4-2

____________ PERIOD _____

Reading to Learn Mathematics Angles of Triangles

Pre-Activity

How are the angles of triangles used to make kites? Read the introduction to Lesson 4-2 at the top of page 185 in your textbook. The frame of the simplest kind of kite divides the kite into four triangles. Describe these four triangles and how they are related to each other.

Reading the Lesson E A

a. Name the three interior angles of the triangle. (Use three letters to name each angle.)

39!

b. Name three exterior angles of the triangle. (Use three letters to name each angle.) c. Name the remote interior angles of "EAB.

D

B

23!

C F

d. Find the measure of each angle without using a protractor. i. "DBC

ii. "ABC

iii. "ACF

iv. "EAB

2. Indicate whether each statement is true or false. If the statement is false, replace the underlined word or number with a word or number that will make the statement true. a. The acute angles of a right triangle are supplementary. b. The sum of the measures of the angles of any triangle is 100. c. A triangle can have at most one right angle or acute angle. d. If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent. e. The measure of an exterior angle of a triangle is equal to the difference of the measures of the two remote interior angles. f. If the measures of two angles of a triangle are 62 and 93, then the measure of the third angle is 35. g. An exterior angle of a triangle forms a linear pair with an interior angle of the triangle.

Helping You Remember 3. Many students remember mathematical ideas and facts more easily if they see them demonstrated visually rather than having them stated in words. Describe a visual way to demonstrate the Angle Sum Theorem.

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Lesson 4-2

1. Refer to the figure.

NAME ______________________________________________ DATE

4-2

____________ PERIOD _____

Enrichment

Finding Angle Measures in Triangles You can use algebra to solve problems involving triangles.

Example

In triangle ABC, m"A, is twice m"B, and m"C is 8 more than m"B. What is the measure of each angle? Write and solve an equation. Let x ! m"B. m"A " m"B " m"C ! 180 2x " x " (x " 8) ! 180 4x " 8 ! 180 4x ! 172 x ! 43 So, m" A ! 2(43) or 86, m"B ! 43, and m"C ! 43 " 8 or 51. Solve each problem. 1. In triangle DEF, m"E is three times m"D, and m"F is 9 less than m"E. What is the measure of each angle?

2. In triangle RST, m"T is 5 more than m"R, and m"S is 10 less than m"T. What is the measure of each angle?

3. In triangle JKL, m"K is four times m"J, and m"L is five times m"J. What is the measure of each angle?

4. In triangle XYZ, m"Z is 2 more than twice m"X, and m"Y is 7 less than twice m"X. What is the measure of each angle?

5. In triangle GHI, m"H is 20 more than m"G, and m"G is 8 more than m"I. What is the measure of each angle?

6. In triangle MNO, m"M is equal to m"N, and m"O is 5 more than three times m"N. What is the measure of each angle?

7. In triangle STU, m"U is half m"T, and m"S is 30 more than m"T. What is the measure of each angle?

8. In triangle PQR, m"P is equal to m"Q, and m"R is 24 less than m"P. What is the measure of each angle?

9. Write your own problems about measures of triangles.

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NAME ______________________________________________ DATE

4-3

____________ PERIOD _____

Study Guide and Intervention Congruent Triangles

Corresponding Parts of Congruent Triangles

S

B

Triangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, !ABC # !RST.

R C

Example

If !XYZ " !RST, name the pairs of congruent angles and congruent sides. "X # "R, "Y # "S, "Z # "T X $Y $#R $S $, X $Z $#R $T $, Y $Z $#S $T $

T

A

Y S X

R

Z

T

Exercises 1.

2.

K

B

B J A

L C

A

C

D

3. K

L

J

M

Name the corresponding congruent angles and sides for the congruent triangles. 4. F

G L

E

©

Glencoe/McGraw-Hill

K J

5. B

6. R

D

U A

C

195

S

T

Glencoe Geometry

Lesson 4-3

Identify the congruent triangles in each figure.

NAME ______________________________________________ DATE

4-3

____________ PERIOD _____

Study Guide and Intervention

(continued)

Congruent Triangles Identify Congruence Transformations

If two triangles are congruent, you can slide, flip, or turn one of the triangles and they will still be congruent. These are called congruence transformations because they do not change the size or shape of the figure. It is common to use prime symbols to distinguish between an original !ABC and a transformed !A$B$C$.

Example

Name the congruence transformation that produces !A#B#C# from !ABC. The congruence transformation is a slide. "A # "A$; "B # "B$; "C #"C$; $B A $#$ A$$$ B$$; A $C $#$ A$$$ C$$; B $C $#$ B$$$ C$$

y

B

B$ O

A

x

C A$

C$

Exercises Describe the congruence transformation between the two triangles as a slide, a flip, or a turn. Then name the congruent triangles. 1.

2.

S y T O

R

3.

y

P

M N$

x

T$

S$

y

N

O

P$

M$

4.

P

y

A

Q O

5.

C

x

Q$

O

P$

6. C

A$

M x

O

B$

P$ A

©

y

N P

x

O

x

B$

B

y

C$

x

Glencoe/McGraw-Hill

N$

B

196

Glencoe Geometry

NAME ______________________________________________ DATE

____________ PERIOD _____

Skills Practice

4-3

Congruent Triangles Identify the congruent triangles in each figure. 1.

P

2.

V

B A

J

X

T L

Y

S

3.

W

4.

Q R

P

C

E F

D

S

G

Name the congruent angles and sides for each pair of congruent triangles. 5. !ABC # !FGH

Verify that each of the following transformations preserves congruence, and name the congruence transformation. 7. !ABC # !A$B$C$

8. !DEF # !D$E$F $

y

B A

©

y

E

B$

E$

x

O

A$

C$

D

C

Glencoe/McGraw-Hill

197

F F$ O

D$

x

Glencoe Geometry

Lesson 4-3

6. !PQR # !STU

NAME ______________________________________________ DATE

____________ PERIOD _____

Practice

4-3

Congruent Triangles Identify the congruent triangles in each figure. 1.

2.

B

M

P N

R A

C S

Q

L

D

Name the congruent angles and sides for each pair of congruent triangles. 3. !GKP # !LMN

4. !ANC # !RBV

Verify that each of the following transformations preserves congruence, and name the congruence transformation. 5. !PST # !P$S$T$

6. !LMN # !L$M$N$

y

S

L

S$ O

P

y

M

T T$

N O

x

L$

P$

x

N$ M$

QUILTING For Exercises 7 and 8, refer to the quilt design.

A

C

D

E

G

F

7. Indicate the triangles that appear to be congruent. B

8. Name the congruent angles and congruent sides of a pair of congruent triangles.

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Glencoe/McGraw-Hill

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I

H

Glencoe Geometry

NAME ______________________________________________ DATE

4-3

____________ PERIOD _____

Reading to Learn Mathematics Congruent Triangles

Pre-Activity

Why are triangles used in bridges? Read the introduction to Lesson 4-3 at the top of page 192 in your textbook. In the bridge shown in the photograph in your textbook, diagonal braces were used to divide squares into two isosceles right triangles. Why do you think these braces are used on the bridge?

Reading the Lesson 1. If !RST # !UWV, complete each pair of congruent parts. "R #

# "W

$T R $#

#U $W $

"T # #W $V $

2. Identify the congruent triangles in each diagram. a.

b.

B

Q

C

A

S

D

P

c. M

R

d. R

Q

T

N

O

S

P

U

3. Determine whether each statement says that congruence of triangles is reflexive, symmetric, or transitive. a. If the first of two triangles is congruent to the second triangle, then the second triangle is congruent to the first. b. If there are three triangles for which the first is congruent to the second and the second is congruent to the third, then the first triangle is congruent to the third. c. Every triangle is congruent to itself.

Helping You Remember 4. A good way to remember something is to explain it to someone else. Your classmate Ben is having trouble writing congruence statements for triangles because he thinks he has to match up three pairs of sides and three pairs of angles. How can you help him understand how to write correct congruence statements more easily?

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Lesson 4-3

V

NAME ______________________________________________ DATE

4-3

____________ PERIOD _____

Enrichment

Transformations in The Coordinate Plane The following statement tells one way to map preimage points to image points in the coordinate plane.

(x, y ) → (x $ 6, y % 3) y

B

(x, y) → (x " 6, y # 3) This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x " 6, y # 3).” With this transformation, for example, (3, 5) is mapped to (3 " 6, 5 # 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.

Y A O

C

x

X Z

1. Does the transformation above appear to be a congruence transformation? Explain your answer. Draw the transformation image for each figure. Then tell whether the transformation is or is not a congruence transformation. 2. (x, y) → (x # 4, y)

3. (x, y) → (x " 8, y " 7)

y

O

y

O

x

%

4. (x, y) → (#x , #y)

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&

1 2

5. (x, y) → # %%x, y

y

O

x

y

x

O

200

x

Glencoe Geometry

NAME ______________________________________________ DATE

4-4

____________ PERIOD _____

Study Guide and Intervention Proving Congruence—SSS, SAS

SSS Postulate You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate lets you show that two triangles are congruent if you know only that the sides of one triangle are congruent to the sides of the second triangle. If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

SSS Postulate

Example

Write a two-column proof. Given: A $B $#D $B $ and C is the midpoint of A $D $. Prove: !ABC # !DBC

B

A

C

Statements

Reasons

$B $#D $B $ 1. A

1. Given

2. C is the midpoint of A $D $.

2. Given

3. A $C $#D $C $

3. Definition of midpoint

4. B $C $#B $C $

4. Reflexive Property of #

5. !ABC # !DBC

5. SSS Postulate

D

Exercises Write a two-column proof. B

A

C

2.

Y

Z

X

R

U

S

$B $#X $Y $, A $C $#X $Z $, B $C $#Y $Z $ Given: A Prove: !ABC # !XYZ

$S $#U $T $, R $T $#U $S $ Given: R Prove: !RST # !UTS

Statements

Reasons

Statements

Reasons

1.A !B !"X !Y !

1.Given

1.R !S !"U !T !

1. Given

2.S !T !"T !S ! 2. Refl. Prop. 3.!RST " !UTS 3. SSS Post.

2.!ABC " !XYZ 2. SSS Post.

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Lesson 4-4

1.

NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-4

(continued)

Proving Congruence—SSS, SAS SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate. SAS Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Example

For each diagram, determine which pairs of triangles can be proved congruent by the SAS Postulate. a. A

b. D

X

B

C

Y

Z

E

F

In !ABC, the angle is not “included” by the sides A $B $ and A $C $. So the triangles cannot be proved congruent by the SAS Postulate.

H

G

J

The right angles are congruent and they are the included angles for the congruent sides. !DEF # !JGH by the SAS Postulate.

c.

P

Q

1 2

S

R

The included angles, "1 and "2, are congruent because they are alternate interior angles for two parallel lines. !PSR # !RQP by the SAS Postulate.

Exercises For each figure, determine which pairs of triangles can be proved congruent by the SAS Postulate. 1.

2.

P T

4.

Q N U

R

V

3. N

X

M

Z

W

5. A

W

Y

B

P

L

M

6. F

G K

M

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T

D

C

202

J

H

Glencoe Geometry

NAME ______________________________________________ DATE

4-4

____________ PERIOD _____

Skills Practice Proving Congruence—SSS, SAS

Determine whether !ABC " !KLM given the coordinates of the vertices. Explain. 1. A(#3, 3), B(#1, 3), C(#3, 1), K(1, 4), L(3, 4), M(1, 6)

2. A(#4, #2), B(#4, 1), C(#1, #1), K(0, #2), L(0, 1), M(4, 1)

3. Write a flow proof. Given: P $R $#D $E $, P $T $#D $F $ "R # "E, "T # "F Prove: !PRT # !DEF

R P

E D

T

F

PR " DE Given

!PRT " !DEF

PT " DF

SAS

Given

"R " "E

"P " "D

Given

Third Angle Theorem

"T " "F

Lesson 4-4

Given

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 4.

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5.

6.

203

Glencoe Geometry

NAME ______________________________________________ DATE

____________ PERIOD _____

Practice

4-4

Proving Congruence—SSS, SAS Determine whether !DEF " !PQR given the coordinates of the vertices. Explain. 1. D(#6, 1), E(1, 2), F(#1, #4), P(0, 5), Q(7, 6), R(5, 0)

2. D(#7, #3), E(#4, #1), F(#2, #5), P(2, #2), Q(5, #4), R(0, #5)

3. Write a flow proof. Given: R $S $#T $S $ V is the midpoint of R $T $. Prove: !RSV # !TSV

R V

S

T

SV " SV RS " TS Given

V is the

midpoint of RT. Given

Reflexive Property

!RSV " !TSV SSS

RV " VT Definition of midpoint

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. 4.

5.

6.

7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in the diagram. How does he know that the lengths A$B$ and AB are equal?

A

B C B$

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A$

Glencoe Geometry

NAME ______________________________________________ DATE

4-4

____________ PERIOD _____

Reading to Learn Mathematics Proving Congruence—SSS, SAS

Pre-Activity

How do land surveyors use congruent triangles? Read the introduction to Lesson 4-4 at the top of page 200 in your textbook. Why do you think that land surveyors would use congruent right triangles rather than other congruent triangles to establish property boundaries?

Reading the Lesson 1. Refer to the figure.

N

a. Name the sides of !LMN for which "L is the included angle. b. Name the sides of !LMN for which "N is the included angle.

L

M

c. Name the sides of !LMN for which "M is the included angle.

2. Determine whether you have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate that you would use. If not, write not possible. a. A

b.

B

E D

D

C

G

d.

G

E

R U

F

S

Lesson 4-4

c. E $H $ and D $G $ bisect each other.

D

F

T

H

Helping You Remember 3. Find three words that explain what it means to say that two triangles are congruent and that can help you recall the meaning of the SSS Postulate.

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NAME ______________________________________________ DATE

4-4

____________ PERIOD _____

Enrichment

Congruent Parts of Regular Polygonal Regions Congruent figures are figures that have exactly the same size and shape. There are many ways to divide regular polygonal regions into congruent parts. Three ways to divide an equilateral triangular region are shown. You can verify that the parts are congruent by tracing one part, then rotating, sliding, or reflecting that part on top of the other parts.

1. Divide each square into four congruent parts. Use three different ways.

2. Divide each pentagon into five congruent parts. Use three different ways.

3. Divide each hexagon into six congruent parts. Use three different ways.

4. What hints might you give another student who is trying to divide figures like those into congruent parts?

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NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-5

Proving Congruence—ASA, AAS ASA Postulate

are congruent.

The Angle-Side-Angle (ASA) Postulate lets you show that two triangles

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

ASA Postulate

Example

Find the missing congruent parts so that the triangles can be proved congruent by the ASA Postulate. Then write the triangle congruence. a.

B

E

A

C D

F

Two pairs of corresponding angles are congruent, "A # "D and "C # "F. If the included sides A $C $ and D $F $ are congruent, then !ABC # !DEF by the ASA Postulate. b. S

X

R

T

W

Y

"R # "Y and S $R $#X $Y $. If "S # "X, then !RST# !YXW by the ASA Postulate.

Exercises What corresponding parts must be congruent in order to prove that the triangles are congruent by the ASA Postulate? Write the triangle congruence statement. 1.

2.

C D

B

E

W

A

Z

5.

B

6.

V R

B

Y

A

4. A

3.

X

E

D

B

C

D

T U

C

A

C

E

Lesson 4-5

D

S

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Glencoe Geometry

NAME ______________________________________________ DATE

4-5

____________ PERIOD _____

Study Guide and Intervention

(continued)

Proving Congruence—ASA, AAS AAS Theorem Another way to show that two triangles are congruent is the AngleAngle-Side (AAS) Theorem. AAS Theorem

If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

You now have five ways to show that two triangles are congruent. • definition of triangle congruence • ASA Postulate • SSS Postulate • AAS Theorem • SAS Postulate

Example

In the diagram, "BCA " "DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Postulate? $C A $#A $C $ by the Reflexive Property of congruence. The congruent angles cannot be "1 and "2, because A $C $ would be the included side. If "B # "D, then !ABC # !ADC by the AAS Theorem.

B A

1 2

C D

Exercises In Exercises 1 and 2, draw and label !ABC and !DEF. Indicate which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Theorem. 1. "A # "D; "B # "E

2. BC # EF; "A # "D

B

C

F

E A

C D

B F

A

3. Write a flow proof. Given: "S # "U; T $R $ bisects "STU. Prove: "SRT # "URT TR bisects "STU. Given

E

D

S R

T

U

"STR " "UTR Def.of " bisector "S " "U

!SRT " !URT

"SRT " "URT

Given

AAS

CPCTC

RT " RT

Refl. Prop. of "

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208

Glencoe Geometry

NAME ______________________________________________ DATE

4-5

____________ PERIOD _____

Skills Practice Proving Congruence—ASA, AAS

Write a flow proof. 1. Given: "N # "L J $K $#M $K $ Prove: !JKN # !MKL

J L

K N M

"N " "L Given

!JKN " !MKL

JK " MK

AAS

Given

"JKN " "MKL Vertical # are ".

2. Given: $ AB $#C $B $ "A # "C D $B $ bisects "ABC. Prove: A $D $#C $D $

B

D

A

C

AB " CB Given

"A " "C

!ABD " !CBD

AD " CD

Given

ASA

CPCTC

DB bisects "ABC.

"ABD " "CBD Def. of " bisector

Given

3. Write a paragraph proof.

F

G

Lesson 4-5

Given: $ DE $ || F $G $ "E # "G Prove: !DFG # !FDE

E

D

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209

Glencoe Geometry

NAME ______________________________________________ DATE

4-5

____________ PERIOD _____

Practice Proving Congruence—ASA, AAS

1. Write a flow proof. Given: S is the midpoint of Q $T $. $R Q $ || T $U $ Prove: !QSR # !TSU S is the midpoint of QT. Given

QR || TU Given

R T S

Q

U

QS " TS Def.of midpoint

"Q " "T Alt. Int. # are ".

!QSR " !TSU ASA

"QSR " "TSU Vertical # are ".

2. Write a paragraph proof. Given: "D # "F G $E $ bisects "DEF. Prove: D $G $#F $G $

D G

E

F

ARCHITECTURE For Exercises 3 and 4, use the following information. An architect used the window design in the diagram when remodeling an art studio. A $B $ and C $B $ each measure 3 feet.

B

A

D

C

3. Suppose D is the midpoint of A $C $. Determine whether !ABD # !CBD. Justify your answer.

4. Suppose "A # "C. Determine whether !ABD # !CBD. Justify your answer.

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210

Glencoe Geometry

NAME ______________________________________________ DATE

4-5

____________ PERIOD _____

Reading to Learn Mathematics Proving Congruence—ASA, AAS

Pre-Activity

How are congruent triangles used in construction? Read the introduction to Lesson 4-5 at the top of page 207 in your textbook. Which of the triangles in the photograph in your textbook appear to be congruent?

Reading the Lesson 1. Explain in your own words the difference between how the ASA Postulate and the AAS Theorem are used to prove that two triangles are congruent.

2. Which of the following conditions are sufficient to prove that two triangles are congruent? A. Two sides of one triangle are congruent to two sides of the other triangle. B. The three sides of one triangles are congruent to the three sides of the other triangle. C. The three angles of one triangle are congruent to the three angles of the other triangle. D. All six corresponding parts of two triangles are congruent. E. Two angles and the included side of one triangle are congruent to two sides and the included angle of the other triangle. F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a nonincluded angle of the other triangle. G. Two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the other triangle. H. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. I. Two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of the other triangle. 3. Determine whether you have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate or theorem that you would use. If not, write not possible. a.

b. T is the midpoint of R $U $.

E

U

S T A

B

C

D R

V

4. A good way to remember mathematical ideas is to summarize them in a general statement. If you want to prove triangles congruent by using three pairs of corresponding parts, what is a good way to remember which combinations of parts will work?

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Glencoe Geometry

Lesson 4-5

Helping You Remember

NAME ______________________________________________ DATE

4-5

____________ PERIOD _____

Enrichment

Congruent Triangles in the Coordinate Plane If you know the coordinates of the vertices of two triangles in the coordinate plane, you can often decide whether the two triangles are congruent. There may be more than one way to do this. 1. Consider ! ABD and !CDB whose vertices have coordinates A(0, 0), B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what you know about congruent triangles and the coordinate plane to show that ! ABD # !CDB. You may wish to make a sketch to help get you started.

2. Consider !PQR and !KLM whose vertices are the following points. P(1, 2) K(#2, 1)

Q(3, 6) L(#6, 3)

R(6, 5) M(#5, 6)

Briefly describe how you can show that !PQR # !KLM.

3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.

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____________ PERIOD _____

Study Guide and Intervention

4-6

Properties of Isosceles Triangles An isosceles triangle has two congruent sides. The angle formed by these sides is called the vertex angle. The other two angles are called base angles. You can prove a theorem and its converse about isosceles triangles. A

• If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem) • If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

B C If A $B $#C $B $, then "A # "C. If "A # "C, then A $B $#C $B $.

Example 1

Example 2

Find x.

Find x.

S

C

(4x $ 5)!

A

(5x % 10)!

B

3x % 13

R

BC ! BA, so m"A ! m"C. 5x # 10 ! 4x " 5 x # 10 ! 5 x ! 15

T

2x

m"S ! m"T, so SR ! TR. 3x # 13 ! 2x 3x ! 2x " 13 x ! 13

Isos. Triangle Theorem Substitution Subtract 4x from each side. Add 10 to each side.

Converse of Isos. ! Thm. Substitution Add 13 to each side. Subtract 2x from each side.

Exercises Find x. 1.

R P

40!

4. D

2x !

2. S

2x $ 6

T

3x % 6

3. V

Q

P

K

T (6x $ 6)!

2x !

Q

5. G

Y

Statements

Glencoe/McGraw-Hill

3x !

6.

B

30!

Z T 3x !

D

3x !

7. Write a two-column proof. Given: "1 # "2 Prove: A $B $#C $B $

©

W

L

R

x!

S

B A

Reasons

213

1

3

C

D

2

E

Glencoe Geometry

Lesson 4-6

Isosceles Triangles

NAME ______________________________________________ DATE

____________ PERIOD _____

Study Guide and Intervention

4-6

(continued)

Isosceles Triangles Properties of Equilateral Triangles

An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem can be used to prove two properties of equilateral triangles. 1. A triangle is equilateral if and only if it is equiangular. 2. Each angle of an equilateral triangle measures 60°.

Example

Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle.

A P 1

Proof:

2 Q

B

C

Statements

Reasons

$Q $ || B $C $. 1. !ABC is equilateral; P 2. m"A ! m"B ! m"C ! 60 3. "1 # "B, "2 # "C 4. m"1 ! 60, m"2 ! 60 5. !APQ is equilateral.

1. Given 2. Each " of an equilateral ! measures 60°. 3. If || lines, then corres. "s are #. 4. Substitution 5. If a ! is equiangular, then it is equilateral.

Exercises Find x. 1.

2.

D

6x !

4.

J

E

P 4x

V

5.

Q 40

60!

L

Y

4x % 4

Glencoe/McGraw-Hill

60! H

4x !

O A D

1 2

B

C

Proof:

©

R M

7. Write a two-column proof. Given: !ABC is equilateral; "1 # "2. Prove: "ADB # "CDB Statements

!KLM is equilateral.

6.

X

Z

K

M

H

3x $ 8 60!

R

3x !

N

5x

6x % 5

F

3. L

G

Reasons

214

Glencoe Geometry

NAME ______________________________________________ DATE

4-6

____________ PERIOD _____

Skills Practice Isosceles Triangles

Refer to the figure.

Lesson 4-6

C

1. If A $C $#A $D $, name two congruent angles. B D

2. If B $E $#B $C $, name two congruent angles.

E

A

3. If "EBA # "EAB, name two congruent segments.

4. If "CED # "CDE, name two congruent segments.

!ABF is isosceles, !CDF is equilateral, and m"AFD " 150. Find each measure. 5. m"CFD

6. m"AFB

7. m"ABF

8. m"A

A

E

F

B L

9. If m"RLP ! 100, find m"BRL. 10. If m"LPR ! 34, find m"B.

R

11. Write a two-column proof. Given: $ CD $#C $G $ $E D $#G $F $ Prove: C $E $#C $F $

Glencoe/McGraw-Hill

D

35!

In the figure, P !L !"R !L ! and L !R !"B !R !.

©

C

B

C

215

P D E F G

Glencoe Geometry

NAME ______________________________________________ DATE

4-6

____________ PERIOD _____

Practice Isosceles Triangles

Refer to the figure.

R

1. If R $V $#R $T $, name two congruent angles.

S V

2. If R $S $#S $V $, name two congruent angles.

T

U

3. If "SRT # "STR, name two congruent segments. 4. If "STV # "SVT, name two congruent segments. Triangles GHM and HJM are isosceles, with G !H !"M !H ! and H !J !"M !J !. Triangle KLM is equilateral, and m"HMK " 50. Find each measure. 5. m"KML

6. m"HMG

7. m"GHM

J

K

L M

H G

8. If m"HJM ! 145, find m"MHJ. 9. If m"G ! 67, find m"GHM. 10. Write a two-column proof. Given: $ DE $ || B $C $ "1 # "2 Prove: A $B $#A $C $

E A

2 1

D

3

4

C

B

11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle.

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n col Lin

ks Haw

Glencoe Geometry

NAME ______________________________________________ DATE

4-6

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

How are triangles used in art? Read the introduction to Lesson 4-6 at the top of page 216 in your textbook. • Why do you think that isosceles and equilateral triangles are used more often than scalene triangles in art? • Why might isosceles right triangles be used in art?

Reading the Lesson 1. Refer to the figure.

R

a. What kind of triangle is !QRS?

S

b. Name the legs of !QRS.

Q

c. Name the base of !QRS. d. Name the vertex angle of !QRS. e. Name the base angles of !QRS.

2. Determine whether each statement is always, sometimes, or never true. a. If a triangle has three congruent sides, then it has three congruent angles. b. If a triangle is isosceles, then it is equilateral. c. If a right triangle is isosceles, then it is equilateral. d. The largest angle of an isosceles triangle is obtuse. e. If a right triangle has a 45° angle, then it is isosceles. f. If an isosceles triangle has three acute angles, then it is equilateral. g. The vertex angle of an isosceles triangle is the largest angle of the triangle. 3. Give the measures of the three angles of each triangle. a. an equilateral triangle b. an isosceles right triangle c. an isosceles triangle in which the measure of the vertex angle is 70 d. an isosceles triangle in which the measure of a base angle is 70 e. an isosceles triangle in which the measure of the vertex angle is twice the measure of one of the base angles

Helping You Remember 4. If a theorem and its converse are both true, you can often remember them most easily by combining them into an “if-and-only-if” statement. Write such a statement for the Isosceles Triangle Theorem and its converse.

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Glencoe Geometry

Lesson 4-6

Isosceles Triangles

NAME ______________________________________________ DATE

____________ PERIOD _____

Enrichment

4-6

Triangle Challenges Some problems include diagrams. If you are not sure how to solve the problem, begin by using the given information. Find the measures of as many angles as you can, writing each measure on the diagram. This may give you more clues to the solution. 2. Given: AC ! AD, and A $B $#B $D $, m"DAC ! 44 and C $E $ bisects " ACD. Find m"DEC.

1. Given: BE ! BF, " BFG # " BEF # "BED, m"BFE ! 82 and ABFG and BCDE each have opposite sides parallel and congruent. Find m" ABC. A

B

A

C

E B G

D F

D

E

3. Given: m"UZY ! 90, m"ZWX ! 45, !YZU # !VWX, UVXY is a square (all sides congruent, all angles right angles). Find m"WZY. U

V

C

4. Given: m"N ! 120, J N#M N, $$ $$ !JNM # !KLM. Find m"JKM. J

W N

K

M L

Z

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218

Glencoe Geometry

NAME ______________________________________________ DATE

4-7

____________ PERIOD _____

Study Guide and Intervention Triangles and Coordinate Proof

Position and Label Triangles A coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the coordinate plane and label the vertices. Use the following guidelines. Use the origin as a vertex or center of the figure. Place at least one side of the polygon on an axis. Keep the figure in the first quadrant if possible. Use coordinates that make the computations as simple as possible.

Example

Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive x-axis. Start with R(0, 0). If RT is a, then another vertex is T(a, 0).

y

a 2

For vertex S, the x-coordinate is %%. Use b for the y-coordinate, a so the vertex is S %%, b . 2

%

&

S #–2a , b$

R (0, 0)

T (a, 0) x

Exercises Find the missing coordinates of each triangle. 1.

y

C (?, q)

A(0, 0) B(2p, 0) x

2.

y

T (?, ?)

R(0, 0) S(2a, 0) x

3.

y

F (?, b)

G(2g, 0) x

E(?, ?)

Position and label each triangle on the coordinate plane. 4. isosceles triangle !RST with base R $S $ 4a units long y

R(0, 0)

©

T(2a, b)

S(4a, 0) x

Glencoe/McGraw-Hill

5. isosceles right !DEF with legs e units long y

D(0, 0)

6. equilateral triangle !EQI with vertex Q(0, a) and sides 2b units long y

F ( e, e)

E(e, 0) x

219

Q(0, a)

E(–b, 0)

I (b, 0) x

Glencoe Geometry

Lesson 4-7

1. 2. 3. 4.

NAME ______________________________________________ DATE

4-7

____________ PERIOD _____

Study Guide and Intervention

(continued)

Triangles and Coordinate Proof Write Coordinate Proofs

Coordinate proofs can be used to prove theorems and to verify properties. Many coordinate proofs use the Distance Formula, Slope Formula, or Midpoint Theorem.

Example

Prove that a segment from the vertex angle of an isosceles triangle to the midpoint of the base is perpendicular to the base. First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(#a, 0), and S(0, c). Then U(0, 0) is the midpoint of R $T $.

y

S(0, c)

R (–a, 0)

U(0, 0) T (a, 0) x

Given: Isosceles !RST; U is the midpoint of base R $T $. Prove: S $U $⊥R $T $ Proof: #a " a 0 " 0 U is the midpoint of R $T $ so the coordinates of U are %%, %% ! (0, 0). Thus S $U $ lies on

%

2

2

&

$T $ lies on the x-axis. The axes are perpendicular, so the y-axis, and !RST was placed so R $U S $⊥R $T $.

Exercises Prove that the segments joining the midpoints of the sides of a right triangle form a right triangle.

B(0, 2b) R A(0, 0)

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Glencoe/McGraw-Hill

220

P Q

C (2a, 0)

Glencoe Geometry

NAME ______________________________________________ DATE

4-7

____________ PERIOD _____

Skills Practice Triangles and Coordinate Proof

Position and label each triangle on the coordinate plane. 2. isosceles !KLP with base K $P $ 6b units long

y

y

H(b, 0) x

y

L(3b, c)

F (0, a)

G(0, 0)

3. isosceles !AND with base A $D $ 5a long

K(0, 0)

N # –25a, b$

P (6b, 0) x

D(5a, 0) x

A(0, 0)

Lesson 4-7

1. right !FGH with legs a units and b units

Find the missing coordinates of each triangle. 4.

5.

y

y

A(0, ?)

C (0, 0) B(2a, 0) x

7.

y

R(2a, b)

X(0, 0)

8.

6.

Z (?, ?)

y

M (?, ?)

Y (2b, 0) x

9.

y

y

T (?, ?)

R(?, ?)

P (0, 0)

Q (?, ?) x

N (0, 0)

P (7b, 0) x

N (3b, 0) x

O (0, 0)

S (–a, 0)

U (a, 0) x

10. Write a coordinate proof to prove that in an isosceles right triangle, the segment from the vertex of the right angle to the midpoint of the hypotenuse is perpendicular to the hypotenuse. Given: isosceles right !ABC with "ABC the right angle and M the midpoint of A $C $ Prove: B $M $⊥A $C $

A(0, 2a) M B (0, 0) C (2a, 0)

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Glencoe/McGraw-Hill

221

Glencoe Geometry

NAME ______________________________________________ DATE

____________ PERIOD _____

Practice

4-7

Triangles and Coordinate Proof Position and label each triangle on the coordinate plane. 1. equilateral !SWY with 1 sides %% a long 4

y

2. isosceles !BLP with base B $L $ 3b units long y

Y # –81a, b$

S(0, 0)

W # –41a, 0$ x

3. isosceles right !DGJ with hypotenuse D $J $ and legs 2a units long y

P # –23b, c$

D (0, 2a)

G(0, 0) J (2a, 0) x

L(3b, 0) x

B(0, 0)

Find the missing coordinates of each triangle. 4.

y

S (?, ?)

5.

6.

y

y

M (0, ?)

E (0, ?)

J (0, 0)

R # –31b, 0$ x

B (–3a, 0)

C (?, 0) x

N (?, 0)

P (2b, 0) x

NEIGHBORHOODS For Exercises 7 and 8, use the following information. Karina lives 6 miles east and 4 miles north of her high school. After school she works part time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school. 7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall are at the vertices of a right triangle. Given: !SKM Prove: !SKM is a right triangle.

y

K (6, 4) M (–2, 3) S (0, 0)

x

8. Find the distance between the mall and Karina’s home.

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222

Glencoe Geometry

NAME ______________________________________________ DATE

____________ PERIOD _____

Reading to Learn Mathematics

4-7

Triangles and Coordinate Proof Pre-Activity

How can the coordinate plane be useful in proofs? Read the introduction to Lesson 4-7 at the top of page 222 in your textbook. From the coordinates of A, B, and C in the drawing in your textbook, what do you know about !ABC?

Lesson 4-7

Reading the Lesson 1. Find the missing coordinates of each triangle. a.

b.

y

R (?, b)

y

F (?, ?)

E (?, a) T(a, ?)

S (?, ?)

x

D (?, ?)

x

2. Refer to the figure.

y

S (0, a)

a. Find the slope of S $R $ and the slope of S $T $. b. Find the product of the slopes of S $R $ and S $T $. What does this tell you about S $R $ and S $T $? c. What does your answer from part b tell you about !RST ?

R (–a, 0)

O (0, 0) T (a, 0) x

d. Find SR and ST. What does this tell you about S $R $ and S $T $? e. What does your answer from part d tell you about !RST? f. Combine your answers from parts c and e to describe !RST as completely as possible.

g. Find m"SRT and m"STR. h. Find m"OSR and m"OST.

Helping You Remember 3. Many students find it easier to remember mathematical formulas if they can put them into words in a compact way. How can you use this approach to remember the slope and midpoint formulas easily?

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NAME ______________________________________________ DATE

____________ PERIOD _____

Enrichment

4-7

How Many Triangles? Each puzzle below contains many triangles. Count them carefully. Some triangles overlap other triangles. How many triangles are there in each figure? 1.

2.

3.

4.

5.

6.

How many triangles can you form by joining points on each circle? List the vertices of each triangle. 7.

8.

B

C

A

8.

F

H

G

I

D

J

E

9.

K

Q

R

P

O L N

©

Glencoe/McGraw-Hill

S

M

V

224

T U

Glencoe Geometry

NAME

DATE

PERIOD

Chapter 4 Test, Form 1

4

SCORE

Write the letter for the correct answer in the blank at the right of each question. 1.

1. How would this triangle be classified by angles? A. acute B. equiangular C. obtuse D. right 2. What is the value of x if !ABC is equilateral? 1 2

C. ""

A

D. 2

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right of each question. 3. What is m"2? A. 50 4. What is m"4? A. 10

6x ! 3

7.5x

10x # 5

C

70" 2

1

60" 3 4

3.

40"

B. 70

C. 110

D. 120 4.

B. 60

C. 100

D. 120

5. What are the congruent triangles in the diagram? A. !ABC ! !EBD B. !ABE ! !CBD C. !AEB ! !CBD D. !ABE ! !CDB 6. If !CJW ! !AGS, m"A # 50, m"J # 45, and m"S # 16x $ 5, what is x? A. 17.5 B. 11.875 C. 6 D. 5

A

5.

C

B

E

D

J

6.

G

45"

C

(16x ! 5)"

A 50"

W

S

7. Which postulate can be used to prove the triangles congruent? A. SSS B. SAS C. ASA D. AAS 8. What reason should be given for statement 5 in the proof? Given: D "B " is the perpendicular bisector of A "C ". Prove: !ADB ! !CDB

A

B

Reasons

1. DB is the perpendicular bisector of A "C ". 2. A "B "!C "B " 3. "ABD ! "CBD 4. D "B "!D "B " 5. !ADB ! !CDB

1. Given 2. Midpoint Theorem 3. ⊥ line; all right # are !. 4. Reflexive Property 5. ?

Glencoe/McGraw-Hill

B. AAS

C. ASA

225

8.

D

Statements

A. SSS ©

7.

C

D. SAS Glencoe Geometry

Assessments

A. !8

2.

B

1 B. !"" 8

NAME

4

DATE

Chapter 4 Test, Form 1

(continued)

Use the proof for Questions 9–10 and write the letter for the correct answer in the blank at the right of each question.

N J

M

L

Given: L is the midpoint of J "M "; J "K " || N "M ". Prove: !JKL ! !MNL Statements Reasons 1. L is the midpoint of J "M ". 2. J "L "!M "L " || 3. J "K " M "N " 4. "JKL ! "MNL 5. "JLK ! "MLN 6. !JKL ! !MNL

PERIOD

K

1. Given 2. Definition of midpoint 3. Given 4. Alt. int. # are !. 5. (Question 9) 6. (Question 10)

9. What is the reason for "JLK ! "MLN? A. definition of midpoint B. corresponding angles C. vertical angles D. alternate interior angles

9.

10. What is the reason for !JKL ! !MNL? A. AAS B. ASA C. SAS

10. D. SSS

Use the figure for Questions 11–12 and write the letter for the correct answer in the blank at the right of each question. 11. If !LMN is isosceles and T is the midpoint of L "N ", which postulate can be used to prove !MLT ! !MNT? A. AAA B. AAS C. SAS

M 12

L

T

11.

N

D. ABC

12. If !MLT ! !MNT, what is used to prove "1 ! "2? A. CPCTC B. definition of isosceles triangle C. definition of perpendicular D. definition of angle bisector 13. What are the missing coordinates of this triangle? A. (2a, 2c) B. (2a, 0) C. (0, 2a) D. (a, 2c)

12.

y

L(0, 0)

Bonus What is the classification by sides of a triangle with coordinates A(5, 0), B(0, 5), and C(!5, 0)? ©

Glencoe/McGraw-Hill

226

13.

M ( a, c)

N(?, ?) x

B:

Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Test, Form 2A

SCORE

Write the letter for the correct answer in the blank at the right of each question. 1. What is the length of the sides of this equilateral triangle? A. 42 B. 30 C. 15 D. 12

9x # 12

3x ! 6 6x # 3

2. What is the classification of !ABC with vertices A(4, 1), B(2, !1), and C(!2, !1) by its sides? A. equilateral B. isosceles C. scalene D. right Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right of each question.

4. What is m"3? A. 40

B. 50

C. 70

2.

70" 2 1

50"

3

3.

D. 90 4.

B. 70

C. 90

D. 110

5. If !DJL ! !EGS, which segment in !EGS corresponds to D "L "? A. E "G " B. E "S " C. G "" S D. G "E " 6. Which triangles are congruent in the figure? A. !KLJ ! !MNL B. !JLK ! !NLM C. !JKL ! !LMN D. !JKL ! !MNL

Reasons

1. R "J " || I"E " 2. "RJN ! "IEN 3. R "I" bisects J "E ". 4. J "N "!E "N " 5. "RNJ ! "INE 6. !RJN ! !IEN

1. Given 2. (Question 7) 3. Given 4. Definition of bisector 5. Vert. # are !. 6. (Question 8)

©

6.

M

K L J

Use the proof for Questions 7–8 and write the letter for the correct answer in the blank at the right of each question. "J " || E "I"; R "I" bisects J "E ". Given: R Prove: !RJN ! !IEN Statements

5.

N R

E

N

J

I

7. What is the reason for statement 2 in the proof? A. Isosceles Triangle Theorem B. same side interior angles C. corresponding angles D. Alternate Interior Angle Theorem

7.

8. What is the reason for statement 6? A. ASA B. AAS

8.

Glencoe/McGraw-Hill

C. SAS

227

D. SSS Glencoe Geometry

Assessments

3. What is m"1? A. 40

1.

NAME

4

DATE

Chapter 4 Test, Form 2A

"E "!F "C ", which theorem 9. If !ABC is isosceles and A or postulate can be used to prove !AEB ! !CFB? A. SSS B. SAS C. ASA D. AAS Use the proof for Questions 10–11 and write the letter for the correct answer in the blank at the right of each question. Given: D "A " || Y "N "; D "A "!Y "N " Prove: "NDY ! "DNA Statements

Reasons

1. D "A " || Y "N " 2. "ADN ! "YND 3. D "A "!Y "N " 4. D "N "!D "N " 5. !NDY ! !DNA 6. "NDY ! "DNA

1. Given 2. Alt. int. # are !. 3. Given 4. Reflexive Property 5. (Question 10) 6. (Question 11)

10. What is the reason for statement 5? A. ASA C. SAS 11. What is the reason for statement 6? A. Alt. int. "s are !. C. Corr. angles are !.

PERIOD

(continued)

9.

B

A

E

C

F

D

A Y

N

10. B. AAS D. SSS 11. B. CPCTC D. Isosceles Triangle Theorem

12. What is the classification of a triangle with vertices A(3, 3), B(6, !2), C(0, !2) 12. by its sides? A. isosceles B. scalene C. equilateral D. right 13. What are the missing coordinates of the triangle? A. (!2b, 0) B. (0, 2b) C. (!c, 0) D. (0, !c)

y (0, c)

(?, ?)

Bonus Name the coordinates of points A and C in isosceles right !ABC if point C is in the second quadrant.

©

Glencoe/McGraw-Hill

228

(2b, 0) x

B:

y

B(0, a)

A(?, ?)

13.

x

Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Test, Form 2B

SCORE

Write the letter for the correct answer in the blank at the right of each question. 1. What is the length of the sides of this equilateral triangle? A. 2.5 B. 5 C. 15 D. 20

1.

3x ! 5

4x

7x # 15

2. What is the classification of !ABC with vertices A(0, 0), B(4, 3), and C(4, !3) by its sides? A. equilateral B. isosceles C. scalene D. right Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right of each question.

4. What is m"2? A. 120

120" 2

3. B. 90

C. 60

D. 30 4.

B. 90

C. 60

D. 30

5. If !TGS ! !KEL, which angle in !KEL corresponds to "T? A. "L B. "E C. "K D. "A 6. Which triangles are congruent in the figure? A. !HMN ! !HGN B. !HMN ! !NGH C. !NMH ! !NGH D. !MNH ! !HGN

M

N

H

G

Use the proof for Questions 7–8 and write the letter for the correct answer in the blank at the right of each question. "B " || C "D "; A "C " bisects B "D ". Given: A Prove: !ABE ! !CDE Statements Reasons 1. A "C " bisects B "D ". 2. B "E "!D "E " 3. A "B " || C "D " 4. "ABE ! "CDE 5. (Question 8) 6. !ABE ! !CDE

8. What is the statement for reason 5? A. "BEA ! "DEC C. "EAB ! "ECD Glencoe/McGraw-Hill

C B

6.

D E A

1. Given 2. (Question 7) 3. Given 4. Alt. int. # are !. 5.Vert. # are !. 5. ASA

7. What is the reason for statement 2? A. Definition of bisector C. Given

©

5.

7. B. Midpoint Theorem D. Alternate Interior Angle Theorem 8. B. "ABE ! "CDE D. "BEC ! "DEA

229

Glencoe Geometry

Assessments

3. What is m"1? A. 120

1

2.

NAME

4

DATE

Chapter 4 Test, Form 2B

9. If A "F "!D "E ", A "B "!F "C " and A "B " || F "C ", which theorem or postulate can be used to prove !ABE ! !FCD? A. AAS B. ASA C. SAS D. SSS Use the proof for Questions 10–11 and write the letter for the correct answer in the blank at the right of each question. Given: E "G " ! I"A "; "EGA ! "IAG Prove: "GEN ! "AIN Statements Reasons 1. E "G " ! I"A "

1. Given

2. "EGA ! "IAG

2. Given

"A "!G "A " 3. G

3. Reflexive Property

4. !EGA ! !IAG

4. (Question 10)

5. "GEN ! "AIN

5. (Question 11)

10. What is the reason for statement 4? A. SSS B. ASA 11. What is the reason for statement 5? A. Alt. int. # are !. C. Corr. angles are !.

PERIOD

(continued)

B

A

9.

C

F

D

E

E

I N G

A

10. C. SAS

D. AAS 11.

B. Same Side Interior Angles D. CPCTC

12. What is the classification of a triangle with vertices A(!3, !1), B(!2, 2), C(3, 1) by its sides? A. scalene B. isosceles C. equilateral D. right

12.

13. What are the missing coordinates of the triangle? A. (a, 0) B. (b, 0) C. (c, 0) D. (0, c)

13.

y (?, ?)

(a, 0) x

(#a, 0)

Bonus Find x in the triangle.

©

Glencoe/McGraw-Hill

(5x ! 60)"

(2x ! 51)"

(43 # 2x)"

(30 # 10x)"

230

B:

Glencoe Geometry

NAME

PERIOD

Chapter 4 Test, Form 2C

SCORE

1. Use a protractor and ruler to classify the triangle by its angles and sides.

1.

2. Find x, AB, BC, and AC if !ABC is equilateral.

2.

B 10x # 6

7x ! 3

A

C

8x

3. Find the measure of the sides of the triangle if the vertices of !EFG are E(!3, 3), F(1, !1), and G(!3, !5). Then classify the triangle by its sides.

Find the measure of each angle.

3.

Assessments

4

DATE

1

4. m"1

4.

2

110"

3

5. m"2

5.

6. m"3

6.

7. Identify the congruent triangles and name their corresponding congruent angles.

D

F

B G

8. Verify that !ABC ! !A%B%C% preserves congruence, assuming that corresponding angles are congruent.

y

7.

C

A

B A

O

8.

C C% x

A% B%

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Glencoe/McGraw-Hill

231

Glencoe Geometry

NAME

DATE

Chapter 4 Test, Form 2C

4

9. ABCD is a quadrilateral with "B A "!C "D " and A "B " || C "D ". Name the postulate that could be used to prove !BAC ! !DCA. Choose from SSS, SAS, ASA, and AAS. 10. !KLM is an isosceles triangle and "1 ! "2. Name the theorem that could be used to determine "LKP ! "LMN. Then name the postulate that could be used to prove !LKP ! !LMN. Choose from SSS, SAS, ASA, and AAS.

(continued)

A

9.

B

D

PERIOD

C

10.

L

1

K

2

P

11. Use the figure to find m"1.

N

M

11.

1 190" 40"

12. Find x.

(18x # 12)"

12.

(10x ! 20)"

15x "

13. Position and label isosceles !ABC with base A "B " b units long on the coordinate plane.

13. C(–2b, c)

A

14. C "P " joins point C in isosceles right !ABC to the midpoint P, of A "B ". Name the coordinates of P. Then determine the relationship between A "B " and C "P ".

14.

y

A(0, b)

C (0, 0)

B(b, 0) x

Bonus Without finding any other angles or sides congruent, which pair of triangles can be proved to be congruent by the HL Theorem? B

A ©

E

C

D

Glencoe/McGraw-Hill

Y

F

X

B(b, 0)

B:

N

Z

M

O

232

Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Test, Form 2D

SCORE

1. Use a protractor and ruler to classify the triangle by its angles and sides.

1.

2. Find x, AB, BC, AC if !ABC is isosceles.

2.

B 5x ! 5

A

2x ! 20

C

9x # 5

3. Find the measure of the sides of the triangle if the vertices of !EFG are E(1, 4), F(5, 1), and G(2, !3). Then classify the triangle by its sides.

3.

4. m"1

4.

80" 1 70"

2

3

5. m"2

5.

6. m"3

6.

7. Identify the congruent triangles and name their corresponding congruent angles.

B

D

C

E

7.

A

8. Verify that !JKL ! !J%K%L% preserves congruence, assuming that corresponding angles are congruent.

Assessments

Find the measure of each angle.

F

y

8.

L%

K

J

x

O

J% K%

L

9. In quadrilateral JKLM, J "K "!L "K " and M "K " bisects "LKJ. Name the postulate that could be used to prove !MKL ! !MKJ. Choose from SSS, SAS, ASA, and AAS.

©

Glencoe/McGraw-Hill

J

9.

M L

K

233

Glencoe Geometry

NAME

DATE

Chapter 4 Test, Form 2D

4

10. !ABC is an isosceles triangle with B "D "⊥A "C ". Name the theorem that could be used to determine "A ! "C. Then name the postulate that could be used to prove !BDA ! !BDC. Choose from SSS, SAS, ASA, and AAS.

(continued)

C B

PERIOD

10.

D A

11. Use the figure to find m"1.

11.

1

80"

12. Find x.

(18x # 8)"

12.

(6x ! 4)"

13. Position and label equilateral !KLM with side lengths 3a units long on the coordinate plane.

13. L(1.5a, b)

K(0, 0)

14. M "N " joins the midpoint of A "B " and the midpoint of A "C " in !ABC. Find the coordinates of M and N, and the slopes of M "N " and B "C ".

14.

y

C(0, b) N(?, ?) A(0, 0)

M(?, ?) B(a, 0) x

Bonus Without finding any other angles or sides congruent, which pair of triangles can be proved to be congruent by the LL Theorem? B

A

©

E

C

D

Glencoe/McGraw-Hill

Y

F

X

M(3a, 0)

B:

N

Z

M

O

234

Glencoe Geometry

NAME

DATE

PERIOD

Chapter 4 Test, Form 3

4

SCORE

1. If !ABC is isosceles, "B is the vertex angle, AB # 20x ! 2, BC # 12x $ 30, and AC # 25x, find x and the measure of each side of the triangle.

1.

2. Given A(0, 4), B(5, 4), and C(!3, !2), find the measure of the sides of the triangle. Then classify the triangle by its sides and angles.

2.

Use the figure to answer Questions 3–5. 1

(3x # 10)"

(8x # 30)"

3. Find x.

3.

4. m"1, if m"1 # 4x $ 10.

4.

5. m"2

5.

6. Verify that the following preserves congruence, assuming that corresponding angles are congruent. !ABC is reflected over the x-axis as follows. A(!1, 1) → A%(!1, !1) B(4, 2) → B%(4, !2) C(1, 5) → C%(1, !5) Verify !ABC ! !A%B%C%.

6.

7. Determine whether !GHI ! !JKL, given G(1, 2), H(5, 4), I(3, 6) and J(!4, !5), K(0, !3), L(!2, !1). Explain.

7.

8. In the figure, A "C "!F "D ", A "B " || D "E ", and A "C " || F "D ". Name the postulate that could be used to prove !ABC ! !DEC. Choose from SSS, SAS, ASA, and AAS.

8.

B

Assessments

2

F D

A C E

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Glencoe/McGraw-Hill

235

Glencoe Geometry

NAME

DATE

Chapter 4 Test, Form 3

4

PERIOD

(continued)

For Questions 9 and 10, complete this two-column proof. Given: !ABC is an isosceles triangle with base A "C ". D is the midpoint of A "C ". Prove: B "D " bisects "ABC.

B 1 2

A

D

Statements

Reasons

1. !ABC is isosceles with base A "C ".

1. Given

"B "!C "B " 2. A

2. Def. of isosceles triangle.

3. "A ! "C

3. (Question 9)

"C ". 4. D is the midpoint of A

4. Given

"D "!C "D " 5. A

5. Midpoint Theorem

6. !ABD ! !CBD

6. (Question 10)

7. "1 ! "2

7. CPCTC

"D " bisects "ABC. 8. B

8. Def. of angle bisector

11. Find x.

C

10.

11.

(17x ! 9)" (21x # 3)"

9.

(15x ! 15)"

12. Position and label isosceles !ABC with base A "B " (a $ b) units long on a coordinate plane

12. C (a !2 b, c)

A(0, 0)

Bonus In the figure, !ABC is isosceles, !ADC is equilateral, !AEC is isosceles, and the measures of "9, "1, and "3 are all equal. Find the measures of the nine numbered angles. A 3

2

C

©

5

B:

1

7 E 4

B(a ! b, 0)

8

D

9

B

6

Glencoe/McGraw-Hill

236

Glencoe Geometry

NAME

4

DATE

Chapter 4 Open-Ended Assessment

PERIOD SCORE

Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem. 1.

(20x # 10)"

(9x ! 4)"

a. Classify the triangle by its angles and sides. b. Show the steps needed to solve for x.

Assessments

2. a. Describe how to determine whether a triangle with coordinates A(1, 4), B(1, !1), and C(!4, 4) is an equilateral triangle. b. Is the triangle equilateral? Explain. 3. Explain how to find m"1 and m"2 in the figure. D 40"

B

E 62"

A 58"

1

2

C

4.

J

G E L

D

S

a. State the theorem or postulate that can be used to prove that the triangles are congruent. b. List their corresponding congruent angles and sides. 5. A

B C E

D

Given: A "B " || D "E ", A "D " bisects B "E ". Prove: !ABC ! !DEC by using the ASA postulate. ©

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Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Vocabulary Test/Review flow proof included angle included side isosceles triangle obtuse triangle

coordinate proof corollary equiangular triangle equilateral triangle exterior angle

acute triangle base angles congruence transformations congruent triangles

SCORE

remote interior angles right triangle scalene triangle vertex angle

Choose from the terms above to complete each sentence. 1. A triangle that is equilateral is also called a(n) 2. A(n)

?

?

.

has at least one obtuse angle.

3. The sum of the triangle. 4. The

?

?

is equivalent to the exterior angle of a

angles of an isosceles triangle are congruent.

1. 2. 3. 4.

5. A triangle with different measures for each side is classified as ? a(n) .

5.

? 6. A organizes a series of statements in logical order written in boxes and uses arrows to indicate the order of the statements.

6.

7. A triangle that is translated, reflected or rotated and preserves ? its shape, is said to be a(n) .

7.

8. The ASA postulate involves two corresponding angles and ? their corresponding .

8.

? 9. A uses figures in the coordinate plane and algebra to prove geometric concepts.

9.

? 10. The triangle.

is formed by the congruent legs of an isosceles

10.

In your own words— 11. corollary

11.

12. congruent triangles

12.

13. acute triangle

13.

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Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Quiz

SCORE

(Lessons 4–1 and 4–2) 1. Use a protractor to classify the triangle by its angles and sides.

1.

2. STANDARDIZED TEST PRACTICE What is the best classification of this triangle by its angles and sides? A. acute isosceles B. right isosceles C. obtuse isosceles D. obtuse equilateral

2.

3.

3. If !ABC is an isosceles triangle, "B is the vertex angle, AB # 6x $ 3, BC # 8x ! 1, and AC # 10x ! 10, find x and the measures of each side of the triangle. 4. If A(1, 5), B(3, !2), and C(!3, 0), find the measures of the sides of !ABC. Then classify the triangle by its sides.

5. m"1

6. m"2

7. m"3

8. m"4

9. m"5

10. m"6

6.

1

7. 70"

65" 5 2 6 107" 4 43"

8.

3

9. 10.

NAME

4

5.

Assessments

Find the measure of each angle in the figure.

4.

DATE

PERIOD

Chapter 4 Quiz

SCORE

(Lessons 4–3 and 4–4) 1. Identify the congruent triangles in the figure.

1.

K N

L

M

2. STANDARDIZED TEST PRACTICE If !JGO ! !RWI, which angle corresponds to "I? A. "J B. "R C. "G D. "O

2.

3. Verify that the following preserves congruence assuming that corresponding angles are congruent. !ABC ! !A%B%C%

3.

B

A%

©

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239

x

O

A

4. In quadrilateral EFGH, F "G "!H "E ", and "G F " || H "E ". Name the postulate that could be used to prove !EHF ! !GFH. Choose from SSS, SAS, ASA, and AAS.

y

B%

C%

C F E

G

4.

H

Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Quiz

SCORE

(Lessons 4–5 and 4–6) For Questions 1 and 2, complete the two-column proof by supplying the missing information for each corresponding location. A Z

Given: "Z ! "C; A "K " bisects "ZKC. Prove: !AKZ ! !AKC Statements Reasons 1. "Z ! "C; A "K " bisects "ZKC. 2. "ZKA ! "CKA 3. A "K "!A "K " 4. !AKZ ! !AKC

K

1. Given 2. (Question 1) 3. Reflexive Property 4. (Question 2)

Refer to the figure for Questions 3 and 4. 3. Find m"1.

C

2. 3.

1 2

4. Find m"2.

4.

NAME

4

1.

DATE

PERIOD

Chapter 4 Quiz

SCORE

(Lesson 4–7) 1. Find the missing coordinates.

y

I (?, ?)

M(#b, 0)

1.

C (?, ?) x

Position and label each triangle on a coordinate plane.

2.

1

"J "; LJ # ""DL and D "L " is 2. Right !DJL with hypotenuse D 2 a units long.

D(0, a) L(0, 0)

1 2

3. isosceles !EGS with base E "S " ""b units long

3.

For Questions 4 and 5, complete the coordinate proof by supplying the missing information for each corresponding location. Given: !ABC with A(!1, 1), B(5, 1), and C(2, 6). Prove: !ABC is isosceles. By the Distance Formula the lengths of the three sides are as follows: (Question 4) . Since (Question 5) , !ABC is isosceles.

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240

J (–2a, 0)

G(–41b, c) E(0, 0) S(–21b, 0)

4.

5. Glencoe Geometry

NAME

4

DATE

PERIOD

Chapter 4 Mid-Chapter Test

SCORE

(Lessons 4–1 through 4–3) Part I Write the letter for the correct answer in the blank at the right of each question. 1. What is the best classification for this triangle? A. acute scalene B. obtuse equilateral C. acute isosceles D. obtuse isosceles

1.

2. What is m"1? A. 50 C. 100 3. What is m"2? A. 40 C. 60

2

B. 60 D. 105

2. 1

60"

50"

50"

B. 50 D. 100

4. If !SJL ! !DMT, which segment in !DMT corresponds to L "S " in !SJL? A. D "T " B. T "D " C. M "D " D. M "" T

3.

4.

Part II 5. Find the measures of the sides of !ABC and classify it by its sides. A(1, 3), B(5, !2), and C(0, !4)

5.

6. In !ABC and !A%B%C%, "A ! "A%, "B ! "B%, and "C ! "C%. Find the lengths needed to prove !ABC ! !A%B%C%.

6.

y

A

B% C% O

x

C B

7. What information would you need to know about P "O " and L "N " for !LMP to be congruent to !NMO by SSS?

A%

7.

N P

M

O

L

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Glencoe Geometry

Assessments

Find the missing angle measures.

NAME

4

DATE

PERIOD

Chapter 4 Cumulative Review

SCORE

(Chapters 1–4) 1. Name the geometric figure that is modeled by the second hand of a clock. (Lesson 1-1)

1.

2. Find the precision for a measurement of 36 inches. (Lesson 1-2)

2.

For Questions 3–5, use the number line. A

B

C

#10 #9 #8 #7 #6 #5 #4 #3 #2 #1

D 0

1

E 2

3

4

5

6

3. Find BC. (Lesson 1-3)

3.

4. Find the coordinate of the midpoint of A "D ". (Lesson 1-3)

4.

5. If B is the midpoint of a segment having one endpoint at E, what is the coordinate of its other endpoint? (Lesson 1-3)

5.

For Questions 6 and 7, determine whether each statement is always, sometimes, or never true. Explain your answer. (Lesson 2-5)

6. If D "E "!E "F ", then E is the midpoint of D "F ". 7. If points A and B lie in plane

6.

Q , then !"# AB lies in Q .

7.

8. Find the slope of a line parallel to x # 2. (Lesson 3-3)

8.

9. Find the distance between y # !9 and y # !5. (Lesson 3-6)

9.

For Questions 10–12, use the figure. 10. Name the segment that represents the distance from F to !"# AD. (Lesson 3-6)

B

A

C

50"

30"

F

85"

D

11. Classify !ADC. (Lesson 4-1)

10.

E

11.

12. Find m"ACD. (Lesson 4-2)

12.

13. Name the corresponding congruent angles and sides for !PQR ! !HGB. (Lesson 4-3)

13.

14. If "QRP ! "SRT, and R is the midpoint of P "T ", which theorem or postulate can be used to prove !QRP ! !SRT? Choose from SSS, SAS, ASA, and AAS. (Lesson 4-5)

Q

15. Name the missing coordinates of !GEF. (Lesson 4-7)

y

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242

14.

T

E(?, ?)

D(0, 0)

©

P

S R

15.

G(?, ?) F (2b, ?) x

Glencoe Geometry

NAME

4

DATE

PERIOD

Standardized Test Practice

SCORE

(Chapters 1–4) Part 1: Multiple Choice Instructions: Fill in the appropriate oval for the best answer.

1.

A

B

C

D

2.

E

F

G

H

3. Complete the statement so that its conditional and its converse are true. ? If "1 ! "2, then "1 and "2 . (Lesson 2-3) A. are supplementary. B. are complementary. C. have the same measure. D. are alternate interior angles.

3.

A

B

C

D

4. Complete this proof. (Lesson 2-7) Given: U "V "!V "W " U "W V "!W "X " Prove: UV # WX Proof: Statements Reasons

4.

E

F

G

H

5.

A

B

C

D

6. Classify !DEF with vertices D(2, 3), E(5, 7) and F(9, 4). (Lesson 4-1) E. acute F. equiangular G. obtuse H. right

6.

E

F

G

H

7. Which postulate or theorem can be used to prove !ABD ! !CBD? (Lesson 4-4) A. SAS B. SSS C. ASA D. AAS

7.

A

B

C

D

1. If m"1 # 5x ! 4, and m"2 # 52 ! 9y, which values for x and y would make "1 and "2 complementary? (Lesson 1-5) A. x # 2, y # 12 B. x # 12, y # 2 1 3

1 3

2. Which is not a polygon? E. F.

D. x # "", y # 27 (Lesson 1-6)

G.

H.

W

X V

"V "!V "W "; V "W "!W "X " 1. U

1. Given

2. UV # VW; VW # WX

2.

3. UV # WX

3. Transitive Property

?

E. Definition of congruent segments F. Substitution Property G. Segment Addition Postulate H. Symmetric Property 1 3

5. Which equation has a slope of "" and a y-intercept of !2? 1 3

A. y # ""x $ 2 1 3

C. y # 2x ! ""

©

Glencoe/McGraw-Hill

(Lesson 3-4)

1 3

B. y # ""x ! 2 1 3

D. y # !2x $ ""

243

A

B

C

D

Glencoe Geometry

Assessments

C. x # 27, y # ""

NAME

4

DATE

Standardized Test Practice

PERIOD

(continued)

Part 2: Grid In Instructions: Enter your answer by writing each digit of the answer in a column box and then shading in the appropriate oval that corresponds to that entry.

8. What is the y-coordinate of the midpoint of A(12, 6) and B(!15, !6)? (Lesson 1-3) 9. If m"1 # 112, find m"10.

1 2 3 4 5 6 7 8 9 10 11 12

L

M

k m n

10.

J

6 4 7

.

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

11.

6

H

5

9.

0

!

(Lesson 3-2)

10. If J "K " || L "M ", then "4 must be supplementary to ? " . (Lesson 3-5)

8.

K

11. Find PR if !PQR is isosceles, "Q is the vertex angle, PQ # 4x ! 8, QR # x $ 7, and PR # 6x ! 12. (Lesson 4-1)

.

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

6 8 .

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

1 8 .

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

Part 3: Short Response Instructions: Show your work or explain in words how you found your answer.

12. The perimeter of a regular pentagon is 14.5 feet. If each side length of the pentagon is doubled, what is the new perimeter?

12.

(Lesson 1-6)

13. Make a conjecture about the next number in the sequence 5, 7, 13. 11, 17, 25. (Lesson 2-1) 14. Find m"PQR. (Lesson 4-2)

14.

Q

P

63"

R

10"

125"

S

T

15. If PQ # QS, QS # SR, and m"R # 20, find m"PSQ. (Lesson 4-6)

©

Glencoe/McGraw-Hill

15.

Q P

S

244

R

Glencoe Geometry

NAME

DATE

PERIOD

Standardized Test Practice

4

Student Record Sheet

(Use with pages 232–233 of the Student Edition.)

Part 1 Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. 1

A

B

C

D

4

A

B

C

D

7

A

B

C

D

2

A

B

C

D

5

A

B

C

D

8

A

B

C

D

3

A

B

C

D

6

A

B

C

D

Part 2 Short Response/Grid In Solve the problem and write your answer in the blank. For Questions 12 and 14, also enter your answer by writing each number or symbol in a box. Then fill in the corresponding oval for that number or symbol. 12

10 11 12

(grid in)

13 14

(grid in)

14 .

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

.

/ .

/ .

.

1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

Answers

9

Part 3 Open-Ended Record your answers for Questions 15–16 on the back of this paper.

©

Glencoe/McGraw-Hill

A1

Glencoe Geometry

© ____________ PERIOD _____

Glencoe/McGraw-Hill

A2

60!

C

Classify each triangle.

35!

25!

F

H

30!

J

The triangle has one right angle. It is a right triangle.

60!

90!

G

The triangle has one angle that is obtuse. It is an obtuse triangle.

D

120!

E

©

65! 65!

acute

U

50!

T

V

23!

Glencoe/McGraw-Hill

4.

90!

67!

right

L

1. K

M

30!

90!

45!

right

X

5. W

P

Y

120!

183

45!

obtuse

2. N 30!

O

6.

3. Q

60!

60!

S

F

obtuse

28!

Glencoe Geometry

92!

D

B 60!

equiangular

R

Classify each triangle as acute, equiangular, obtuse, or right.

60!

All three angles are congruent, so all three angles have measure 60°. The triangle is an equiangular triangle.

B

A

Exercises

c.

b.

a.

Example

• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.

• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.

• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.

• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.

of its angles.

One way to classify a triangle is by the measures

Classifying Triangles

Study Guide and Intervention

Classify Triangles by Angles

4-1

NAME ______________________________________________ DATE

L

J

R

P

All three sides are congruent. The triangle is an equilateral triangle.

N

©

!" 3

W

isosceles

S

scalene

G

2

U

C

1

A

K

18

G 18

I

32x

isosceles

C

8x

32x

equilateral

18

5. B

2.

A

X

15

23

V

T 12

M

19

F

x

equilateral

x

x

E

17

scalene

Q

12

O

The triangle has no pair of congruent sides. It is a scalene triangle.

6. D

3.

c.

Glencoe/McGraw-Hill

184

AB " BC " %65 !, AC " %130 !; !ABC is isosceles. Glencoe Geometry

9. Find the measure of each side of !ABC with vertices A(#1, 5), B(6, 1), and C(2, #6). Classify the triangle.

AB " BC " 8, AC " 6

8. Find the measure of each side of isosceles !ABC with AB ! BC if AB ! 4y, BC ! 3y " 2, and AC ! 3y.

7. Find the measure of each side of equilateral !RST with RS ! 2x " 2, ST ! 3x, and TR ! 5x # 4. 2

4.

1.

Classify each triangle as equilateral, isosceles, or scalene.

b.

Classify each triangle.

Two sides are congruent. The triangle is an isosceles triangle.

H

Example

Exercises

a.

• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.

• If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.

• If all three sides of a triangle are congruent, then the triangle is an equilateral triangle.

You can classify a triangle by the measures of its sides. Equal numbers of hash marks indicate congruent sides.

Classifying Triangles

(continued)

____________ PERIOD _____

Study Guide and Intervention

Classify Triangles by Sides

4-1

NAME ______________________________________________ DATE

Answers (Lesson 4-1)

Glencoe Geometry

Lesson 4-1

©

Classifying Triangles

Skills Practice

____________ PERIOD _____

Glencoe/McGraw-Hill

acute

equiangular 5.

2.

obtuse

obtuse

A3 !BDE

10. obtuse

!BCD, !BDE

8. isosceles E

A

6.

3.

acute

right

D

B

C

©

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185

RS " 5, ST " %10 !, RT " %17 !; scalene

14. R(1, 3), S(4, 7), T(5, 4)

RS " %13 !, ST " %13 !, RT " 4; isosceles

13. R(0, 2), S(2, 5), T(4, 2)

Glencoe Geometry

Answers

Glencoe Geometry

Find the measures of the sides of !RST and classify each triangle by its sides.

x " 4, DE " 11, DF " 11, EF " 13

12. !DEF is isosceles, "D is the vertex angle, DE ! x " 7, DF ! 3x # 1, and EF ! 2x " 5.

x " 6, AB " 16, BC " 16, CA " 16

11. !ABC is equilateral with AB! 3x # 2, BC ! 2x " 4, and CA ! x " 10.

ALGEBRA Find x and the measure of each side of the triangle.

!ABE, !BCE

9. scalene

!ABE, !BCE

7. right

Identify the indicated type of triangles.

4.

1.

Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

4-1

NAME ______________________________________________ DATE

(Average)

Classifying Triangles

Practice

obtuse

2.

acute

!ABD, !BED, !BDC

7. isosceles

!BED, !BDC

5. obtuse

3.

A

right

B

D

E

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186

13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor. How many right angles are there? 5

KP " 2%10 !, PL " 5%2 !, LK " 5%2 !; isosceles

12. K(#2, #6), P(#4, 0), L(3, #1)

KP " %53 !, PL " 5, LK " 2%13 !; scalene

11. K(5, #3), P(3, 4), L(#1, 1)

KP " %26 !, PL " 4%2 !, LK " %26 !; isosceles

10. K(#3, 2) P(2, 1), L(#2, #3)

Glencoe Geometry

Find the measures of the sides of !KPL and classify each triangle by its sides.

x " 3, LM " 7, LN " 7, MN " 13

9. !LMN is isosceles, "L is the vertex angle, LM ! 3x # 2, LN ! 2x " 1, and MN ! 5x # 2.

x " 7, FG " 12, GH " 12, FH " 12

8. !FGH is equilateral with FG ! x " 5, GH ! 3x # 9, and FH ! 2x # 2.

ALGEBRA Find x and the measure of each side of the triangle.

!ABC, !CDE

6. scalene

!ABC, !CDE

4. right

Identify the indicated type of triangles if A !B !"A !D !"B !D !"D !C !, ! BE !"E !D !, ! AB !⊥B !C !, and ! ED !⊥! DC !.

1.

C

____________ PERIOD _____

Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

4-1

NAME ______________________________________________ DATE

Answers (Lesson 4-1)

Lesson 4-1

©

Glencoe/McGraw-Hill

A4

triangles are symmetrical.

• Why do you think that isosceles triangles are used more often than scalene triangles in construction? Sample answer: Isosceles

Sample answer: Triangles lie in a plane and are rigid shapes.

• Why are triangles used for braces in construction rather than other shapes?

Read the introduction to Lesson 4-1 at the top of page 178 in your textbook.

Why are triangles important in construction?

Classifying Triangles

1 right angle(s), and

0 right angle(s), and

3 acute angle(s), 2 acute angle(s),

0 right angle(s), and

2 acute angle(s),

70!

30!

acute, scalene

80!

b.

obtuse, isosceles

135!

c. 5

3

right, scalene

4

Glencoe/McGraw-Hill

187

Glencoe Geometry

as sharp. An acute pain is a sharp pain, and an acute angle can be thought of as an angle with a sharp point. In an acute triangle all of the angles are acute.

4. A good way to remember a new mathematical term is to relate it to a nonmathematical definition of the same word. How is the use of the word acute, when used to describe acute pain, related to the use of the word acute when used to describe an acute angle or an acute triangle? Sample answer: Both are related to the meaning of acute

Helping You Remember

a.

3. Describe each triangle by as many of the following words as apply: acute, obtuse, right, scalene, isosceles, or equilateral.

f. A scalene triangle is obtuse. sometimes

e. An acute triangle is isosceles. sometimes

d. An equilateral triangle is isosceles. always

c. An equilateral triangle is a right triangle. never

b. An obtuse triangle is isosceles. sometimes

a. A right triangle is scalene. sometimes

2. Determine whether each statement is always, sometimes, or never true.

0 obtuse angle(s).

c. In a right triangle, there are

0 obtuse angle(s).

b. In an acute triangle, there are

1 obtuse angle(s).

a. In an obtuse triangle, there are

1. Supply the correct numbers to complete each sentence.

Reading the Lesson

©

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

4-1

NAME ______________________________________________ DATE

Enrichment

A O

y

B

©

Glencoe/McGraw-Hill

3. Consider three noncollinear points, D, E, and F on a coordinate grid. The x-coordinates of D and E are opposites. The y-coordinates of D and E are the same. The x-coordinate of F is 0. What kind of triangle must !DEF be: scalene, isosceles, or equilateral? isosceles

1. Consider three points, R, S, and T on a coordinate grid. The x-coordinates of R and S are the same. The y-coordinate of T is between the y-coordinates of R and S. The x-coordinate of T is less than the x-coordinate of R. Is angle R of triangle RST acute, right, or obtuse? acute

188

x

Glencoe Geometry

4. Consider three points, G, H, and I on a coordinate grid. Points G and H are on the positive y-axis, and the y-coordinate of G is twice the y-coordinate of H. Point I is on the positive x-axis, and the x-coordinate of I is greater than the y-coordinate of G. Is triangle GHI scalene, isosceles, or equilateral? scalene

2. Consider three noncollinear points, J, K, and L on a coordinate grid. The y-coordinates of J and K are the same. The x-coordinates of K and L are the same. Is triangle JKL acute, right, or obtuse? right

Answer each question. Draw a simple triangle on the grid above to help you.

From the diagram you can see that triangle ABC must be obtuse.

Side AB must be a horizontal segment because the y-coordinates are the same. Point C must be located to the right and up from point B.

To answer this question, first draw a sample triangle that fits the description.

Example Consider three points, A, B, and C on a coordinate grid. The y-coordinates of A and B are the same. The x-coordinate of B is greater than the x-coordinate of A. Both coordinates of C are greater than the corresponding coordinates of B. Is triangle ABC acute, right, or obtuse? Q

____________ PERIOD _____

When you read geometry, you may need to draw a diagram to make the text easier to understand.

Reading Mathematics

4-1

NAME ______________________________________________ DATE

Answers (Lesson 4-1)

Glencoe Geometry

Lesson 4-1

©

Angles of Triangles

Study Guide and Intervention

____________ PERIOD _____

Glencoe/McGraw-Hill

T

35!

A5

Subtract 60 from each side.

Add.

Substitution

Angle Sum Theorem

©

P

1 2

R

1 30!

T 60! W

2

60!

1

T

N

M

m"1 " 28

30!

S

m"1 " 30, m"2 " 60

m"1 " 30, m"2 " 60

90!

62!

Glencoe/McGraw-Hill

5.

U

W

3. V

1.

A

B C

58! 1

C 2

E

108!

189

6. A

P

4. M

Q

58!

66!

Q

152!

1

G

20!

30!

2

1

S

Glencoe Geometry

O

180 180 180 40 Subtract 140 from each side.

Add.

Substitution

Angle Sum Theorem

D

Glencoe Geometry

m"1 " 8

m"1 " 56, m"2 " 56, m"3 " 74

R

m"1 " 120

! ! ! !

Vertical angles are congruent.

Subtract 148 from each side.

Add.

Substitution

Angle Sum Theorem

Answers

1

50!

3

N

m"3 " m"2 " m"E m"3 " 32 " 108 m"3 " 140 m"3

2.

180 180 180 32

D

m"2 ! 32

! ! ! !

3

m"1 " m"A " m"B m"1 " 58 " 90 m"1 " 148 m"1

A

90!

B

Find the missing angle measures.

Example 2

Find the measure of each numbered angle.

Exercises

25 " 35 " m"T ! 180 60 " m"T ! 180 m"T ! 120

m"R " m"S " m"T ! 180

R

25!

S

Find m"T.

The sum of the measures of the angles of a triangle is 180. In the figure at the right, m"A " m"B " m"C ! 180.

Example 1

Angle Sum Theorem

Angle Sum Theorem If the measures of two angles of a triangle are known, the measure of the third angle can always be found.

4-2

NAME ______________________________________________ DATE

T

S

Exercises

Add.

Substitution

©

W

3

O

1

N

2

60!

P

60!

B

2x !

A 95!

Glencoe/McGraw-Hill

5.

1

Z

M

145!

C

D

25

R

25!

2 1

C

35!

D

A

B

Subtract 55 from each side.

Substitution

2 35!

1

V

3 36!

S

T

H

58!

G

x! x!

F

29

Glencoe Geometry

m"1 " 109, m"2 " 29, m"3 " 71

U

80!

R

A

Exterior Angle Theorem

m"1 " 60, m"2 " 120

B

6. E

4.

2.

190

m"1 " 60, m"2 " 60, m"3 " 120

Q

Find x.

3.

65!

m"1 " 115

Y

50!

X

55!

1

C

Find x.

D

m"PQS ! m"R " m"S 78 ! 55 " x 23 ! x

Find the measure of each numbered angle. 1.

x!

78! Q

80! 60!

P

Exterior Angle Theorem

Example 2

S 1

Find m"1.

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m"1 ! m"A " m"B

m"1 ! m"R " m"S ! 60 " 80 ! 140

R

Example 1

Exterior Angle Theorem

At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, "B and "A are the remote interior angles for exterior "DCB.

Angles of Triangles

(continued)

____________ PERIOD _____

Study Guide and Intervention

Exterior Angle Theorem

4-2

NAME ______________________________________________ DATE

Answers (Lesson 4-2)

Lesson 4-2

©

Angles of Triangles

Skills Practice

Glencoe/McGraw-Hill

73!

S TIGER

80!

27

A6

©

Glencoe/McGraw-Hill

15. m"2 27

14. m"1 27

Find the measure of each angle.

13. m"5 115

12. m"4 75

11. m"3 65

10. m"2 40

9. m"1 140

Find the measure of each angle.

8. m"3 95

7. m"2 55

6. m"1 125

Find the measure of each angle.

5. m"3 70

4. m"2 55

3. m"1 55

Find the measure of each angle.

1.

Find the missing angle measures.

4-2

191

2.

A

60!

2

80!

1

40!

70!

85!

146!

55!

1

NAME ______________________________________________ DATE

1 2

2

2

3

D

1

B

3

4 105!

40! 5

150!

Glencoe Geometry

63! C

3

55!

17, 17

____________ PERIOD _____

(Average)

Angles of Triangles

Practice

72! ?

18 2.

©

Glencoe/McGraw-Hill

55 192

14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridge construction. Use the diagram to find m"1.

13. m"2 32

12. m"1 26

Find the measure of each angle if "BAD and "BDC are right angles and m"ABC " 84.

11. m"6 147

10. m"5 73

9. m"2 79

8. m"3 65

7. m"4 45

6. m"1 104

Find the measure of each angle.

5. m"3 62

4. m"2 83

3. m"1 97

Find the measure of each angle.

1.

Find the missing angle measures.

4-2

A

1

39!

58!

B

40!

1

68!

2

36!

1

55!

NAME ______________________________________________ DATE

2

3

35!

3

D

1

5

145!

118! 6

Glencoe Geometry

82!

4 65!

70!

64! C

2

85

____________ PERIOD _____

Answers (Lesson 4-2)

Glencoe Geometry

Lesson 4-2

©

Glencoe/McGraw-Hill

A7

ii. "ABC 118

iii. "ACF 157

39!

iv. "EAB 141

A B

23!

D

C F

Glencoe/McGraw-Hill

193

Glencoe Geometry

Answers

Glencoe Geometry

Sample answer: Cut off the angles of a triangle and place them side-by-side on one side of a line so that their vertices meet at a common point. The result will show three angles whose measures add up to 180.

3. Many students remember mathematical ideas and facts more easily if they see them demonstrated visually rather than having them stated in words. Describe a visual way to demonstrate the Angle Sum Theorem.

Helping You Remember

g. An exterior angle of a triangle forms a linear pair with an interior angle of the triangle. true

f. If the measures of two angles of a triangle are 62 and 93, then the measure of the third angle is 35. false; 25

e. The measure of an exterior angle of a triangle is equal to the difference of the measures of the two remote interior angles. false; sum

d. If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent. true

c. A triangle can have at most one right angle or acute angle. false; obtuse

b. The sum of the measures of the angles of any triangle is 100. false; 180

a. The acute angles of a right triangle are supplementary. false; complementary

2. Indicate whether each statement is true or false. If the statement is false, replace the underlined word or number with a word or number that will make the statement true.

i. "DBC 62

d. Find the measure of each angle without using a protractor.

c. Name the remote interior angles of "EAB. "ABC, "BCA

b. Name three exterior angles of the triangle. (Use three letters to name each angle.) "EAB, "DBC, "FCA

a. Name the three interior angles of the triangle. (Use three letters to name each angle.) "BAC, "ABC, "BCA

1. Refer to the figure. E

Sample answer: There are two pairs of right triangles that have the same size and shape.

The frame of the simplest kind of kite divides the kite into four triangles. Describe these four triangles and how they are related to each other.

Read the introduction to Lesson 4-2 at the top of page 185 in your textbook.

How are the angles of triangles used to make kites?

Angles of Triangles

Reading the Lesson

©

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

4-2

NAME ______________________________________________ DATE

Enrichment

____________ PERIOD _____

©

Glencoe/McGraw-Hill

See students’ work.

194

Glencoe Geometry

m"P " m"Q " 68, m"R " 44

8. In triangle PQR, m"P is equal to m"Q, and m"R is 24 less than m"P. What is the measure of each angle?

m"M " m"N " 35, m"O " 110

6. In triangle MNO, m"M is equal to m"N, and m"O is 5 more than three times m"N. What is the measure of each angle?

m"X " 37, m"Y " 67, m"Z " 76

4. In triangle XYZ, m"Z is 2 more than twice m"X, and m"Y is 7 less than twice m"X. What is the measure of each angle?

m"R " 60, m"S " 55, m"T " 65

2. In triangle RST, m"T is 5 more than m"R, and m"S is 10 less than m"T. What is the measure of each angle?

9. Write your own problems about measures of triangles.

m"S " 90, m"T " 60, m"U " 30

7. In triangle STU, m"U is half m"T, and m"S is 30 more than m"T. What is the measure of each angle?

m"G " 56, m"H " 76, m"I " 48

5. In triangle GHI, m"H is 20 more than m"G, and m"G is 8 more than m"I. What is the measure of each angle?

m"J " 18, m"K " 72, m"L " 90

3. In triangle JKL, m"K is four times m"J, and m"L is five times m"J. What is the measure of each angle?

m"D " 27, m"E " 81, m"F " 72

1. In triangle DEF, m"E is three times m"D, and m"F is 9 less than m"E. What is the measure of each angle?

Solve each problem.

So, m" A ! 2(43) or 86, m"B ! 43, and m"C ! 43 " 8 or 51.

m"A " m"B " m"C ! 180 2x " x " (x " 8) ! 180 4x " 8 ! 180 4x ! 172 x ! 43

Write and solve an equation. Let x ! m"B.

Example In triangle ABC, m"A, is twice m"B, and m"C is 8 more than m"B. What is the measure of each angle?

You can use algebra to solve problems involving triangles.

Finding Angle Measures in Triangles

4-2

NAME ______________________________________________ DATE

Answers (Lesson 4-2)

Lesson 4-2

©

Congruent Triangles

Glencoe/McGraw-Hill

A8

J

C

K

!ABC " !JKL

A

B

L

2.

C

!ABC " !DCB

A

B D

X Z

Y S

M

L

T

A

!JKM " !LMK

J

3. K

C

R

R

T

S

©

G L

J

K

Glencoe/McGraw-Hill

"E " "J; "F " "K; "G " "L; E !F !"J !K !; !G E !"J !L !; F !G !"K !L !

E

4. F A

D C

195

"A " "D ; "ABC " "DCB; "ACB " "DBC ; !B A !"D !C !; A !C !"D !B !; !C B !"C !B !

5. B S

Glencoe Geometry

"R " "T; "RSU " "TSU; "RUS " "TUS; !U R !"T !U !; R !S !"T !S !; !U S !"S !U !

T

U

6. R

Name the corresponding congruent angles and sides for the congruent triangles.

1.

Identify the congruent triangles in each figure.

Exercises

Example If !XYZ " !RST, name the pairs of congruent angles and congruent sides. "X # "R, "Y # "S, "Z # "T XY $ $#$ RS $, $ XZ $#$ RT $, $ YZ $#$ ST $

Triangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, !ABC # !RST.

B

____________ PERIOD _____

Study Guide and Intervention

Corresponding Parts of Congruent Triangles

4-3

NAME ______________________________________________ DATE

A

B

C

O

y

A$

©

O

S$

T$

T x

Q$

Q

P

P$

x

B$

O

A

y

B

C x

slide; !ABC " !A#B #C #

A$

C$

turn; !OPQ " !OP #Q #

O

y

flip; !RST " !RS #T #

R

S y

Glencoe/McGraw-Hill

5.

3.

1.

196

6.

4.

2.

O

y

P$

P x

C B$

x

O

M

y

N$

x

P$

turn; !MNP " !MN #P #

P

N

flip; !ABC " !AB #C

B

O

A

y

Glencoe Geometry

slide; !MNP " !M #N #P #

M$

M N$

N

x

C$

B$

Describe the congruence transformation between the two triangles as a slide, a flip, or a turn. Then name the congruent triangles.

Exercises

Example Name the congruence transformation that produces !A#B#C# from !ABC. The congruence transformation is a slide. "A # "A$; "B # "B$; "C #"C$; AB $ $#$ A$$$ B$$; $ AC $#$ A$$$ C$$; $ BC $#$ B$$$ C$$

If two triangles are congruent, you can slide, flip, or turn one of the triangles and they will still be congruent. These are called congruence transformations because they do not change the size or shape of the figure. It is common to use prime symbols to distinguish between an original !ABC and a transformed !A$B$C$.

Congruent Triangles

(continued)

____________ PERIOD _____

Study Guide and Intervention

Identify Congruence Transformations

4-3

NAME ______________________________________________ DATE

Answers (Lesson 4-3)

Glencoe Geometry

Lesson 4-3

© X

A

W

C

(Average)

Glencoe/McGraw-Hill

L

T

S

S

R

!PQR " !PSR

P

Q

!JPL " !TVS

J

P

V

4.

2.

G

F

!DEF " !DGF

D

E

!ABC " !WXY

Y

A9

©

Glencoe/McGraw-Hill

Glencoe Geometry

Answers

Glencoe Geometry

"F " "F #; flip

"B " "B #, "C " "C #; slide

x

"D " "D #, "E " "E #,

D$

AC " 4, A#C # " 4, "A " "A#,

197

O

F F$

E$

E #F # " 5, DF " 3, D #F # " 3,

D

y

DE " 4, D #E # " 4, EF " 5,

C$

x

E

BC " 2%2 !, B #C # " 2%2 !,

C

A$

B$

8. !DEF # !D$E$F$

AB " 2%2 !, A#B # " 2%2 !,

A

B

O

y

7. !ABC # !A$B$C$

Verify that each of the following transformations preserves congruence, and name the congruence transformation.

"P " "S, "Q " "T, "R " "U; P !Q !"S !T !, Q !R !"T !U !, P !R !"S !U !

6. !PQR # !STU

"A " "F, "B " "G, "C " "H; A !B !"F !G !, B !C !"G !H !, A !C !"F !H !

5. !ABC # !FGH

Name the congruent angles and sides for each pair of congruent triangles.

3.

1. C S

R

!ABC " !DRS

A

B

D

2. N

P

!LMN " !QPN

L

M

Q

____________ PERIOD _____

©

Glencoe/McGraw-Hill

198

Sample answer: "A " "E, "ABI " "EBF, "I " "F; !B A !"E !B !, B !I! " B !F !, A !I! " E !F !

8. Name the congruent angles and congruent sides of a pair of congruent triangles.

!ABI " !EBF, !CBD " !HBG

7. Indicate the triangles that appear to be congruent.

H

B

C

G

D

F

E

Glencoe Geometry

I

A

"M " "M #, "N " "N #; flip QUILTING For Exercises 7 and 8, refer to the quilt design.

LN " 7, L#N # " 7, "L " "L#,

N$

x

N

"S " "S #, "T " !T #; flip

M$

O

MN " %29 !, M #N # " %29 !,

L$

L

P #T # " %10 !, "P " "P #,

P$

x

LM " 2%2 !, L#M # " 2%2 !,

S$

y

ST " %5 !, S #T # " %5 !, PT " %10 !,

T T$

O

M

6. !LMN # !L$M$N$

PS " %13 !, P #S # " %13 !,

P

S

y

5. !PST # !P$S$T$

Verify that each of the following transformations preserves congruence, and name the congruence transformation.

"A " "R, "N " "B, "C " "V ; A !N !"R !B !, N !C !"B !V !, A !C !"R !V !

4. !ANC # !RBV

"G " "L, "K " "M, "P " "N ; G !K !"L !M !, K !P !"M !N !, G !P !"L !N !

3. !GKP # !LMN

Name the congruent angles and sides for each pair of congruent triangles.

1.

Identify the congruent triangles in each figure. B

Practice

Identify the congruent triangles in each figure.

4-3

NAME ______________________________________________ DATE

Congruent Triangles

Skills Practice

____________ PERIOD _____

Lesson 4-3

Congruent Triangles

4-3

NAME ______________________________________________ DATE

Answers (Lesson 4-3)

©

Glencoe/McGraw-Hill "S # "W $W $ !S R ! #U

"R # "U

!V U !

A

A10

N

O

D

C

B

P

V

T

R

U

!PQS " !RQS

c. Every triangle is congruent to itself. reflexive

b. If there are three triangles for which the first is congruent to the second and the second is congruent to the third, then the first triangle is congruent to the third. transitive

a. If the first of two triangles is congruent to the second triangle, then the second triangle is congruent to the first. symmetric

Glencoe/McGraw-Hill

199

Glencoe Geometry

three vertices of one triangle in any order. Then write the corresponding vertices of the second triangle in the same order. If the angles are written in the correct correspondence, the sides will automatically be in the correct correspondence also.

4. A good way to remember something is to explain it to someone else. Your classmate Ben is having trouble writing congruence statements for triangles because he thinks he has to match up three pairs of sides and three pairs of angles. How can you help him understand how to write correct congruence statements more easily? Sample answer: Write the

Helping You Remember

©

S

S

Q

!RTV " !USV

d. R

b.

WV $ !T S ! #$

3. Determine whether each statement says that congruence of triangles is reflexive, symmetric, or transitive.

Q

!ABC " !ADC

!MNO " !QPO

c. M

a.

2. Identify the congruent triangles in each diagram.

P

1. If !RST # !UWV, complete each pair of congruent parts. "T # "V

diagonal braces make the structure stronger and prevent it from being deformed when it has to withstand a heavy load.

In the bridge shown in the photograph in your textbook, diagonal braces were used to divide squares into two isosceles right triangles. Why do you think these braces are used on the bridge? Sample answer: The

Read the introduction to Lesson 4-3 at the top of page 192 in your textbook.

Why are triangles used in bridges?

Congruent Triangles

Reading the Lesson

$ $# RT

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

4-3

NAME ______________________________________________ DATE

Enrichment

C

X

O

©

Glencoe/McGraw-Hill

O

y

4. (x, y) → (#x , #y) yes

O

y

2. (x, y) → (x # 4, y) yes

x

x

%

200

O

1 2

y

& 5. (x, y) → # %%x, y

O

y

no

x

x

3. (x, y) → (x " 8, y " 7) yes

Z

Y x

Glencoe Geometry

Draw the transformation image for each figure. Then tell whether the transformation is or is not a congruence transformation.

changing its size or shape.

A

B

(x, y ) → (x $ 6, y % 3) y

____________ PERIOD _____

1. Does the transformation above appear to be a congruence transformation? Explain your answer. Yes; the transformation slides the figure to the lower right without

This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x " 6, y # 3).” With this transformation, for example, (3, 5) is mapped to (3 " 6, 5 # 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.

(x, y) → (x " 6, y # 3)

The following statement tells one way to map preimage points to image points in the coordinate plane.

Transformations in The Coordinate Plane

4-3

NAME ______________________________________________ DATE

Answers (Lesson 4-3)

Glencoe Geometry

Lesson 4-3

©

Proving Congruence—SSS, SAS

Study Guide and Intervention

____________ PERIOD _____

Glencoe/McGraw-Hill

A11

©

X

201

R

S

1. A !B !"X !Y ! 1.Given !C A !"X !Z ! !C B !"Y !Z ! 2.!ABC " !XYZ 2. SSS Post.

Reasons

Glencoe Geometry

Answers

Glencoe Geometry

1. R !S !"U !T ! 1. Given !T R !"U !S ! 2. S !T !"T !S ! 2. Refl. Prop. 3.!RST " !UTS 3. SSS Post.

Reasons

Statements

Z

Statements

C

U

$#$ UT $, $ RT $#$ US $ Given: $ RS Prove: !RST # !UTS

A

Y

D

$#$ XY $, $ AC $#$ XZ $, $ BC $#$ YZ $ Given: $ AB Prove: !ABC # !XYZ

B

Glencoe/McGraw-Hill

1.

Write a two-column proof. T

5. SSS Postulate

5. !ABC # !DBC

2.

4. Reflexive Property of #

$C $#$ BC $ 4. B

Exercises

2. Given 3. Definition of midpoint

C

$#$ DC $ 3. $ AC

1. Given

$B $#$ DB $ 1. A

A

$D $. 2. C is the midpoint of A

Reasons

Statements

B

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

Write a two-column proof. Given: A $B $#$ DB $ and C is the midpoint of A $D $. Prove: !ABC # !DBC

Example

SSS Postulate

SSS Postulate You know that two triangles are congruent if corresponding sides are congruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate lets you show that two triangles are congruent if you know only that the sides of one triangle are congruent to the sides of the second triangle.

4-4

NAME ______________________________________________ DATE

Proving Congruence—SSS, SAS

Study Guide and Intervention

(continued)

____________ PERIOD _____

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

C

Y

X Z

G

E

J

H

The right angles are congruent and they are the included angles for the congruent sides. !DEF # !JGH by the SAS Postulate.

F

b. D

c.

1 2

R

Q

The included angles, "1 and "2, are congruent because they are alternate interior angles for two parallel lines. !PSR # !RQP by the SAS Postulate.

S

P

©

R

N U

P M

W

T

The triangles cannot be proved congruent by the SAS Postulate.

M

V

!TRU " !PMN by the SAS Postulate.

T

Glencoe/McGraw-Hill

4.

1.

W

X

Z

Y

C

D

202

"D " "B because both are right angles. The two triangles are congruent by the SAS Postulate.

B

"XQY and "WQZ are not the included angles for the congruent segments. The triangles are not congruent by the SAS Postulate.

Q

5. A

2.

L

J

K H

G

Glencoe Geometry

The congruent angles are the included angles for the congruent sides. !FJH " !GHJ by the SAS Postulate.

6. F

"MPL " "NPL because both are right angles. !MPL " !NPL by the SAS Postulate.

M

P

3. N

For each figure, determine which pairs of triangles can be proved congruent by the SAS Postulate.

Exercises

In !ABC, the angle is not “included” by the sides $ AB $ and A $C $. So the triangles cannot be proved congruent by the SAS Postulate.

B

a. A

Example For each diagram, determine which pairs of triangles can be proved congruent by the SAS Postulate.

SAS Postulate

SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.

4-4

NAME ______________________________________________ DATE

Answers (Lesson 4-4)

Lesson 4-4

©

Proving Congruence—SSS, SAS

Skills Practice

____________ PERIOD _____

Glencoe/McGraw-Hill

A12

Third Angle Theorem

Given

SAS

!PRT " !DEF

P T

R D F

E

©

SSS

Glencoe/McGraw-Hill

4.

5.

SAS 203

6.

Glencoe Geometry

not possible

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Given

"T " "F

"P " "D

"R " "E

Given

PT " DF

Given

PR " DE

Proof:

3. Write a flow proof. Given: P $R $#$ DE $, $ PT $#$ DF $ "R # "E, "T # "F Prove: !PRT # !DEF

The corresponding sides are not congruent, so !ABC is not congruent to !KLM.

AB " 3, KL " 3, BC " %13 !, LM " 4, AC " %10 !, KM " 5.

2. A(#4, #2), B(#4, 1), C(#1, #1), K(0, #2), L(0, 1), M(4, 1)

The corresponding sides have the same measure and are congruent, so !ABC " !KLM by SSS.

AB " 2, KL " 2, BC " 2%2 !, LM " 2%2 !, AC " 2, KM " 2.

1. A(#3, 3), B(#1, 3), C(#3, 1), K(1, 4), L(3, 4), M(1, 6)

Determine whether !ABC " !KLM given the coordinates of the vertices. Explain.

4-4

NAME ______________________________________________ DATE

(Average)

Proving Congruence—SSS, SAS

Practice

____________ PERIOD _____

SSS

!RSV " !TSV

Reflexive Property

SV " SV

R

T

V

S

©

not possible

5.

SAS or SSS

6.

SSS A C

B

Glencoe/McGraw-Hill

204

Glencoe Geometry

Since "ACB and "A#CB# are vertical angles, they are A$ B$ congruent. In the figure, A !C !"A !#!C ! and B !C !"! B !#! C. So !ABC " !A#B #C by SAS. By CPCTC, the lengths A#B # and AB are equal.

7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in the diagram. How does he know that the lengths A$B$ and AB are equal?

4.

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Definition of midpoint

RV " VT

V is the midpoint of RT. Given

Given

RS " TS

Proof:

3. Write a flow proof. Given: R $S $#$ TS $ V is the midpoint of R $T $. Prove: !RSV # !TSV

DE " %! 13, PQ " %13, ! EF " 2%5 !, QR " %26 !, DF " %29 !, PR " %13 !. Corresponding sides are not congruent, so !DEF is not congruent to !PQR.

2. D(#7, #3), E(#4, #1), F(#2, #5), P(2, #2), Q(5, #4), R(0, #5)

DE " 5%2 !, PQ " 5%2 !, EF " 2%10 !, QR " 2%10 !, DF " 5%2 !, PR " 5%2 !. !DEF " !PQR by SSS since corresponding sides have the same measure and are congruent.

1. D(#6, 1), E(1, 2), F(#1, #4), P(0, 5), Q(7, 6), R(5, 0)

Determine whether !DEF " !PQR given the coordinates of the vertices. Explain.

4-4

NAME ______________________________________________ DATE

Answers (Lesson 4-4)

Glencoe Geometry

Lesson 4-4

©

Glencoe/McGraw-Hill

A13

L

M

D

F

H

G

F

S T

!RSU # !TSU ; SSS

U

R

not possible

D

E

Glencoe/McGraw-Hill

205

Glencoe Geometry

Answers

Glencoe Geometry

Sample answer: Congruent triangles are triangles that are the same size and shape, and the SSS Postulate ensures that two triangles with three corresponding sides congruent will be the same size and shape.

d.

b.

3. Find three words that explain what it means to say that two triangles are congruent and that can help you recall the meaning of the SSS Postulate.

Helping You Remember

!DEF " !GHF; SAS

D

E

G

c. E $H $ and D $G $ bisect each other.

!ABD " !CBD ; SAS

C

B

a. A

2. Determine whether you have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate that you would use. If not, write not possible.

!L M !, M !N !

c. Name the sides of !LMN for which "M is the included angle.

!L N !, N !M !

b. Name the sides of !LMN for which "N is the included angle.

!M L !, L !N !

a. Name the sides of !LMN for which "L is the included angle.

1. Refer to the figure. N

Sample answer: Land is usually divided into rectangular lots, so their boundaries meet at right angles.

Why do you think that land surveyors would use congruent right triangles rather than other congruent triangles to establish property boundaries?

Read the introduction to Lesson 4-4 at the top of page 200 in your textbook.

How do land surveyors use congruent triangles?

Proving Congruence—SSS, SAS

Reading the Lesson

©

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

4-4

NAME ______________________________________________ DATE

Enrichment

____________ PERIOD _____

©

Glencoe/McGraw-Hill

206

4. What hints might you give another student who is trying to divide figures like those into congruent parts? See students’ work.

3. Divide each hexagon into six congruent parts. Use three different ways. Sample answers are shown.

2. Divide each pentagon into five congruent parts. Use three different ways. Sample answers are shown.

1. Divide each square into four congruent parts. Use three different ways. Sample answers are shown.

Glencoe Geometry

Congruent figures are figures that have exactly the same size and shape. There are many ways to divide regular polygonal regions into congruent parts. Three ways to divide an equilateral triangular region are shown. You can verify that the parts are congruent by tracing one part, then rotating, sliding, or reflecting that part on top of the other parts.

Congruent Parts of Regular Polygonal Regions

4-4

NAME ______________________________________________ DATE

Answers (Lesson 4-4)

Lesson 4-4

© ____________ PERIOD _____

Glencoe/McGraw-Hill

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

C D

E

F

W

Y

T

R

A14

©

5.

Y

C

Glencoe/McGraw-Hill

!D B !"D !B !; "ADB " "CBD; !ABD " !CDB

D

B T U

207

ST ! !"V !T !; !RST " !UVT

S

R

V

Z

W

X

WY ! !"W !Y !; "XYW " "ZYW ; !WXY " !WZY

B

2.

A

E

C

DC ! !"B !C !; !CDE " !CBA

D

4. A

1.

6.

3.

E

D

C

C

D

E

Glencoe Geometry

"ACB " "E; !ABC " !CDE

A

B

"ABE " "CBD ; !ABE " !CBD

A

B

What corresponding parts must be congruent in order to prove that the triangles are congruent by the ASA Postulate? Write the triangle congruence statement.

Exercises

$#$ XY $. If "S # "X, then !RST# !YXW by the ASA Postulate. "R # "Y and $ SR

X

Two pairs of corresponding angles are congruent, "A # "D and "C # "F. If the $ and D $F $ are congruent, then !ABC # !DEF by the ASA Postulate. included sides $ AC

A

B

b. S

a.

Find the missing congruent parts so that the triangles can be proved congruent by the ASA Postulate. Then write the triangle congruence.

Example

ASA Postulate

are congruent.

The Angle-Side-Angle (ASA) Postulate lets you show that two triangles

Proving Congruence—ASA, AAS

Study Guide and Intervention

ASA Postulate

4-5

NAME ______________________________________________ DATE

1 2

D

B C

©

D

C

E F

Glencoe/McGraw-Hill

Given

TR bisects "STU.

Refl. Prop. of "

208

CPCTC

RT " RT

"SRT " "URT AAS

T

!SRT " !URT

R

E

Given

U

S

D

B

F

Glencoe Geometry

If "C " "F (or if "B " "E ), then !ABC " !DEF by the AAS Theorem.

A

C

2. BC # EF; "A # "D

"S " "U

"STR " "UTR Def.of " bisector

3. Write a flow proof. Given: "S # "U; $ TR $ bisects "STU. Prove: "SRT # "URT

If B !C !"E !F ! (or if A !C !"D !F ! ), then !ABC " !DEF by the AAS Theorem.

A

B

1. "A # "D; "B # "E

In Exercises 1 and 2, draw and label !ABC and !DEF. Indicate which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Theorem.

Exercises

Example In the diagram, "BCA " "DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Postulate? A $C $#$ AC $ by the Reflexive Property of congruence. The congruent angles cannot be "1 and "2, because $ AC $ would be the included side. If "B # "D, then !ABC # !ADC by the AAS Theorem.

A

If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

You now have five ways to show that two triangles are congruent. • definition of triangle congruence • ASA Postulate • SSS Postulate • AAS Theorem • SAS Postulate

AAS Theorem

Another way to show that two triangles are congruent is the AngleAngle-Side (AAS) Theorem.

Proving Congruence—ASA, AAS

(continued)

____________ PERIOD _____

Study Guide and Intervention

AAS Theorem

4-5

NAME ______________________________________________ DATE

Answers (Lesson 4-5)

Glencoe Geometry

Lesson 4-5

©

Glencoe/McGraw-Hill

A15

©

AAS

G

"ABD " "CBD Def. of " bisector

DB bisects "ABC.

F

C

M

L

____________ PERIOD _____

Glencoe/McGraw-Hill

209

Glencoe Geometry

Answers

Glencoe Geometry

Proof: Since it is given that D !E ! || F !G !, it follows that "EDF " "GFD, because alt. int. # are ". It is given that "E " "G. By the Reflexive Property, D !F !"F !D !. So !DFG " !FDE by AAS.

$E $ || F $G $ Given: D "E # "G Prove: !DFG # !FDE

3. Write a paragraph proof. E

CPCTC

ASA

Given

AD " CD

!ABD " !CBD

D

Given

D

A

B

K

"A " "C

Given

AB " CB

Proof:

N

!JKN " !MKL

2. Given: A $B $#$ CB $ "A # "C DB $ $ bisects "ABC. Prove: A $D $#$ CD $

"JKN " "MKL Vertical # are ".

Given

JK " MK

Given

"N " "L

Proof:

1. Given: "N # "L JK $ $#$ MK $ Prove: !JKN # !MKL J

Proving Congruence—ASA, AAS

Skills Practice

Write a flow proof.

4-5

NAME ______________________________________________ DATE

(Average)

midpoint of QT.

"QSR " "TSU Vertical # are ".

"Q " "T Alt. Int. # are ".

Def.of midpoint

QS " TS

F

D

U

G

ASA

!QSR " !TSU

S

T

E

©

A

D

B

Glencoe/McGraw-Hill

210

Glencoe Geometry

We are given A !B !"C !B ! and "A " "C. B !D !"B !D ! by the Reflexive Property. Since SSA cannot be used to prove that triangles are congruent, we cannot say whether !ABD " !CBD.

4. Suppose "A # "C. Determine whether !ABD # !CBD. Justify your answer.

Since D is the midpoint of A !C !, A !D !"C !D ! by the definition of midpoint. !B A !"C !B ! by the definition of congruent segments. By the Reflexive Property, B !D !"B !D !. So !ABD " !CBD by SSS.

3. Suppose D is the midpoint of A $C $. Determine whether !ABD # !CBD. Justify your answer.

ARCHITECTURE For Exercises 3 and 4, use the following information. An architect used the window design in the diagram when remodeling an art studio. A $B $ and C $B $ each measure 3 feet.

C

____________ PERIOD _____

Proof: Since it is given that G !E ! bisects "DEF, "DEG " "FEG by the definition of an angle bisector. It is given that "D " "F. By the Reflexive Property, G !E !"G !E !. So !DEG " !FEG by AAS. Therefore !G D !"F !G ! by CPCTC.

Given: "D # "F GE $ $ bisects "DEF. Prove: D $G $#$ FG $

2. Write a paragraph proof.

Given

QR || TU

Given

S is the

Sample proof:

Q

R

Proving Congruence—ASA, AAS

Practice

1. Write a flow proof. Given: S is the midpoint of Q $T $. Q $R $ || T $U $ Prove: !QSR # !TSU

4-5

NAME ______________________________________________ DATE

Answers (Lesson 4-5)

Lesson 4-5

©

Glencoe/McGraw-Hill

A16

to each other. The two obtuse isosceles triangles are congruent to each other.

How are congruent triangles used in construction? Read the introduction to Lesson 4-5 at the top of page 207 in your textbook. Which of the triangles in the photograph in your textbook appear to be congruent? Sample answer: The four right triangles are congruent

Proving Congruence—ASA, AAS

A

B

E

C

D

!AEB " !DEC; AAS

R

S T V

U

!RST " !UVT; ASA

b. T is the midpoint of R $U $.

Glencoe/McGraw-Hill

211

Glencoe Geometry

Sample answer: At least one pair of corresponding parts must be sides. If you use two pairs of sides and one pair of angles, the angles must be the included angles. If you use two pairs of angles and one pair of sides, then the sides must both be included by the angles or must both be corresponding nonincluded sides.

4. A good way to remember mathematical ideas is to summarize them in a general statement. If you want to prove triangles congruent by using three pairs of corresponding parts, what is a good way to remember which combinations of parts will work?

Helping You Remember

a.

3. Determine whether you have enough information to prove that the two triangles in each figure are congruent. If so, write a congruence statement and name the congruence postulate or theorem that you would use. If not, write not possible.

2. Which of the following conditions are sufficient to prove that two triangles are congruent? A. Two sides of one triangle are congruent to two sides of the other triangle. B. The three sides of one triangles are congruent to the three sides of the other triangle. C. The three angles of one triangle are congruent to the three angles of the other triangle. D. All six corresponding parts of two triangles are congruent. E. Two angles and the included side of one triangle are congruent to two sides and the included angle of the other triangle. F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a nonincluded angle of the other triangle. G. Two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the other triangle. H. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. I. Two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of the other triangle.

Sample answer: In ASA, you use two pairs of congruent angles and the included congruent sides. In AAS, you use two pairs of congruent angles and a pair of nonincluded congruent sides. B, D, E, G, H

1. Explain in your own words the difference between how the ASA Postulate and the AAS Theorem are used to prove that two triangles are congruent.

Reading the Lesson

©

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

4-5

NAME ______________________________________________ DATE

Enrichment

____________ PERIOD _____

©

Q(3, 6) L(#6, 3)

R(6, 5) M(#5, 6)

Glencoe/McGraw-Hill

212

Yes; you can use the Distance Formula and SSS.

3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.

Use the Distance Formula to find the lengths of the sides of both triangles. Conclude that ! PQR " ! KLM by SSS.

Briefly describe how you can show that !PQR # !KLM.

P(1, 2) K(#2, 1)

2. Consider !PQR and !KLM whose vertices are the following points.

Glencoe Geometry

Sample answer: Show that the slopes of A !B ! and C !D ! are equal and that the slopes of A !D ! and B !C ! are equal. Conclude A! B&! C! D and ! B! C&! A! D . Use the angle relationships for that ! parallel lines and a transversal and the fact that B !D ! is a common side for the triangles to conclude that !ABD " !CDB by ASA.

1. Consider ! ABD and !CDB whose vertices have coordinates A(0, 0), B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what you know about congruent triangles and the coordinate plane to show that ! ABD # !CDB. You may wish to make a sketch to help get you started.

If you know the coordinates of the vertices of two triangles in the coordinate plane, you can often decide whether the two triangles are congruent. There may be more than one way to do this.

Congruent Triangles in the Coordinate Plane

4-5

NAME ______________________________________________ DATE

Answers (Lesson 4-5)

Glencoe Geometry

Lesson 4-5

© ____________ PERIOD _____

Isosceles Triangles

Study Guide and Intervention

Glencoe/McGraw-Hill

(5x % 10)!

B

(4x $ 5)!

A17

©

P

40!

Q

P

35

2x !

Q

K

2x !

R

12

D

Glencoe/McGraw-Hill

213

36

Glencoe Geometry

Answers

Glencoe Geometry

3.Transitive Property of "

E

S

4.If two angles of a triangle are ", then the sides opposite the angles are ".

2

T 3x !

Z

!B !"C !B ! 4. A

C

R

x!

3x !

15

3."1 " "3

3

6.

Y

W

2.Vertical angles are congruent.

1

B

20

3.

1.Given

A

L

12

Subtract 2x from each side.

Add 13 to each side.

Substitution

Converse of Isos. ! Thm.

2."2 " "3

D

B 3x !

V

T

1."1 " "2

30!

3x % 6

2x $ 6

2x

Reasons

5. G

T

2. S

R

If "A # "C, then A $B $#C $B $.

Find x.

m"S ! m"T, so SR ! TR. 3x # 13 ! 2x 3x ! 2x " 13 x ! 13

3x % 13

S

Example 2

B If A $B $#C $B $, then "A # "C.

C

A

Statements

7. Write a two-column proof. Given: "1 # "2 Prove: $ AB $#$ CB $

T (6x $ 6)!

4. D

1.

Find x.

Exercises

Add 10 to each side.

Subtract 4x from each side.

Substitution

Isos. Triangle Theorem

Find x.

BC ! BA, so m"A ! m"C. 5x # 10 ! 4x " 5 x # 10 ! 5 x ! 15

A

C

Example 1

• If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem) • If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

An isosceles triangle has two congruent sides. The angle formed by these sides is called the vertex angle. The other two angles are called base angles. You can prove a theorem and its converse about isosceles triangles.

Properties of Isosceles Triangles

4-6

NAME ______________________________________________ DATE

©

V

F

4x 60!

P

6x !

D

L

E Q 40

10

R

10

5.

2.

X

G

4x % 4

3x $ 8 60!

J

6x % 5

Z

P 1

2

A B

M

4x !

O

R 60! H

15

10

Glencoe/McGraw-Hill

214

Glencoe Geometry

1. Given 2. An equilateral ! has " sides and " angles. 3. Given 4. ASA Postulate 5. CPCTC

1 2

6.

!KLM is equilateral.

K

3x !

1. !ABC is equilateral. 2. A !B !"C !B ! ; "A " "C 3. "1 " "2 4. !ABD " !CBD 5. "ADB " "CDB

C

D

12

M

N

3. L

Reasons

Y

H

5x

5

C

Q

Statements

Proof:

B

A

1. Given 2. Each " of an equilateral ! measures 60°. 3. If || lines, then corres. "s are #. 4. Substitution 5. If a ! is equiangular, then it is equilateral.

Reasons

7. Write a two-column proof. Given: !ABC is equilateral; "1 # "2. Prove: "ADB # "CDB

4.

1.

Find x.

Exercises

$ || B $C $. 1. !ABC is equilateral; $ PQ 2. m"A ! m"B ! m"C ! 60 3. "1 # "B, "2 # "C 4. m"1 ! 60, m"2 ! 60 5. !APQ is equilateral.

Statements

Proof:

Example Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle.

1. A triangle is equilateral if and only if it is equiangular. 2. Each angle of an equilateral triangle measures 60°.

An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem can be used to prove two properties of equilateral triangles.

Isosceles Triangles

(continued)

____________ PERIOD _____

Study Guide and Intervention

Properties of Equilateral Triangles

4-6

NAME ______________________________________________ DATE

Answers (Lesson 4-6)

Lesson 4-6

©

Isosceles Triangles

Skills Practice

Glencoe/McGraw-Hill 8. m"A 55

7. m"ABF 70

A18

©

4. SAS 5. CPCTC

4. !CDE " !CGF

5. C !E !"C !F !

Glencoe/McGraw-Hill

3. Given

3. D !E !"G !F !

215

E

D

Glencoe Geometry

2. If 2 sides of a ! are ", then the # opposite those sides are ".

2. "D " "G

F G

D E

P

Reasons

L

F

35!

C

1. Given

B

B

E

D

1. C !D !"C !G !

C

R

A

A

B

C

____________ PERIOD _____

Statements

Proof:

CG $#$ $ Given: $ CD $E D $#$ GF $ Prove: C $E $#$ CF $

11. Write a two-column proof.

10. If m"LPR ! 34, find m"B. 68

9. If m"RLP ! 100, find m"BRL. 20

In the figure, P !L !"R !L ! and L !R !"B !R !.

6. m"AFB 55

5. m"CFD 60

!ABF is isosceles, !CDF is equilateral, and m"AFD " 150. Find each measure.

!E C !"C !D !

4. If "CED # "CDE, name two congruent segments.

!B E !"E !A !

3. If "EBA # "EAB, name two congruent segments.

"BEC " "BCE

2. If $ BE $#$ BC $, name two congruent angles.

"ACD " "CDA

1. If A $C $#$ AD $, name two congruent angles.

Refer to the figure.

4-6

NAME ______________________________________________ DATE

(Average)

Isosceles Triangles

Practice

6. m"HMG 70

D

1

4

3

B

C

©

Glencoe/McGraw-Hill

H

J

U

S

K

216

L in

co l

aw

ks

T

L

G

M

Glencoe Geometry

nH

5. If 2 # of a ! are ", then the sides opposite those # are ".

4. Congruence of # is transitive.

3. Given

2. Corr. # are ".

1. Given

Reasons

A

2

E

7. m"GHM 40

11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle.

81, 81

5. A !B !"A !C !

4. "3 " "4

3. "1 " "2

2. "1 " "4 "2 " "3

1. D !E ! || B !C !

Statements

Proof:

$ || B $C $ Given: $ DE "1 # "2 Prove: A $B $#$ AC $

10. Write a two-column proof.

9. If m"G ! 67, find m"GHM. 46

8. If m"HJM ! 145, find m"MHJ. 17.5

5. m"KML 60

V

R

____________ PERIOD _____

Triangles GHM and HJM are isosceles, with G !H !"M !H ! and H !J !"M !J !. Triangle KLM is equilateral, and m"HMK " 50. Find each measure.

4. If "STV # "SVT, name two congruent segments. ! ST !"! SV !

3. If "SRT # "STR, name two congruent segments. ! ST !"! SR !

2. If $ RS $#$ SV $, name two congruent angles. "SVR " "SRV

1. If R $V $#$ RT $, name two congruent angles. "RTV " "RVT

Refer to the figure.

4-6

NAME ______________________________________________ DATE

Answers (Lesson 4-6)

Glencoe Geometry

Lesson 4-6

©

Glencoe/McGraw-Hill

A19

Q

S

Glencoe/McGraw-Hill

217

Glencoe Geometry

Answers

Glencoe Geometry

congruent if and only if the angles opposite those sides are congruent.

4. If a theorem and its converse are both true, you can often remember them most easily by combining them into an “if-and-only-if” statement. Write such a statement for the Isosceles Triangle Theorem and its converse. Sample answer: Two sides of a triangle are

Helping You Remember

e. an isosceles triangle in which the measure of the vertex angle is twice the measure of one of the base angles 90, 45, 45

d. an isosceles triangle in which the measure of a base angle is 70 70, 70, 40

c. an isosceles triangle in which the measure of the vertex angle is 70 70, 55, 55

b. an isosceles right triangle 45, 45, 90

a. an equilateral triangle 60, 60, 60

3. Give the measures of the three angles of each triangle.

sometimes

g. The vertex angle of an isosceles triangle is the largest angle of the triangle.

f. If an isosceles triangle has three acute angles, then it is equilateral. sometimes

e. If a right triangle has a 45° angle, then it is isosceles. always

d. The largest angle of an isosceles triangle is obtuse. sometimes

c. If a right triangle is isosceles, then it is equilateral. never

b. If a triangle is isosceles, then it is equilateral. sometimes

a. If a triangle has three congruent sides, then it has three congruent angles. always

2. Determine whether each statement is always, sometimes, or never true.

e. Name the base angles of !QRS. "Q, "R

d. Name the vertex angle of !QRS. "S

c. Name the base of !QRS. ! QR !

b. Name the legs of !QRS. ! QS !, ! RS !

a. What kind of triangle is !QRS? isosceles

1. Refer to the figure. R

Two congruent isosceles right triangles can be placed together to form a square.

• Why might isosceles right triangles be used in art? Sample answer:

symmetry is pleasing to the eye.

• Why do you think that isosceles and equilateral triangles are used more often than scalene triangles in art? Sample answer: Their

Read the introduction to Lesson 4-6 at the top of page 216 in your textbook.

How are triangles used in art?

Isosceles Triangles

Reading the Lesson

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Reading to Learn Mathematics

Pre-Activity

4-6

NAME ______________________________________________ DATE

Enrichment

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Glencoe/McGraw-Hill

Z

U

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W

3. Given: m"UZY ! 90, m"ZWX ! 45, !YZU # !VWX, UVXY is a square (all sides congruent, all angles right angles). Find m"WZY. 45

G

A

1. Given: BE ! BF, " BFG # " BEF # "BED, m"BFE ! 82 and ABFG and BCDE each have opposite sides parallel and congruent. Find m" ABC. 148

218

B C

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4. Given: m"N ! 120, J N#M $$ $N $, !JNM # !KLM. Find m"JKM. 15

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2. Given: AC ! AD, and A $B $#B $D $, m"DAC ! 44 and C $E $ bisects " ACD. Find m"DEC. 78

K

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Some problems include diagrams. If you are not sure how to solve the problem, begin by using the given information. Find the measures of as many angles as you can, writing each measure on the diagram. This may give you more clues to the solution.

Triangle Challenges

4-6

NAME ______________________________________________ DATE

Answers (Lesson 4-6)

Lesson 4-6

© ____________ PERIOD _____

Triangles and Coordinate Proof

Study Guide and Intervention

Glencoe/McGraw-Hill

Use the origin as a vertex or center of the figure. Place at least one side of the polygon on an axis. Keep the figure in the first quadrant if possible. Use coordinates that make the computations as simple as possible.

Exercises

&

A20

C (?, q)

C (p, q)

A(0, 0) B(2p, 0) x

y

2. T (?, ?)

T (2a, 2a)

R(0, 0) S(2a, 0) x

y

©

S(4a, 0) x

T(2a, b)

Glencoe/McGraw-Hill

R(0, 0)

y

4. isosceles triangle !RST with base $ RS $ 4a units long

D(0, 0)

y

219

E(e, 0) x

F (e, e)

5. isosceles right !DEF with legs e units long

G(2g, 0) x

E (%2g, 0); F (0, b)

E(?, ?)

F (?, b)

y

T (a, 0) x

S #a–2, b$

R (0, 0)

y

Sample answers

3.

E(–b, 0)

Glencoe Geometry

I (b, 0) x

Q(0, a)

y

6. equilateral triangle !EQI with vertex Q(0, a) and sides 2b units long

Position and label each triangle on the coordinate plane. are given.

1.

Find the missing coordinates of each triangle.

%

a so the vertex is S %%, b . 2

For vertex S, the x-coordinate is %%. Use b for the y-coordinate,

a 2

Example Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive x-axis. Start with R(0, 0). If RT is a, then another vertex is T(a, 0).

1. 2. 3. 4.

A coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the coordinate plane and label the vertices. Use the following guidelines.

Position and Label Triangles

4-7

NAME ______________________________________________ DATE

2

2

&

U(0, 0) T (a, 0) x

2b $ 0 2

$

0$0 2

0 $ 2b 2

# 0 $2 2a

# 0 $2 0

$

y

Q

P C (2a, 0) x

0 a b &&, which is undefined, so the segment is vertical. 0

A(0, 0)

R

B(0, 2b)

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Glencoe/McGraw-Hill

220

Glencoe Geometry

"RPQ is a right angle because any horizontal line is perpendicular to any vertical line. !PRQ has a right angle, so !PRQ is a right triangle.

!P ! is && " && " 0, so the segment is horizontal. The slope of R

b%b a%0 b%0 !! Q is && " The slope of P a%a

The midpoint R of AB is &&, && " (0, b).

$

The midpoint Q of AC is &&, && " (a, 0).

The midpoint P of BC is &&, && " (a, b).

# 0 $2 2a

Sample answer: Position and label right !ABC with the coordinates A(0, 0), B(0, 2b), and C (2a, 0).

Prove that the segments joining the midpoints of the sides of a right triangle form a right triangle.

Exercises

$ lies on the x-axis. The axes are perpendicular, so the y-axis, and !RST was placed so $ RT SU $ $⊥$ RT $.

%

R (–a, 0)

y

S(0, c)

Proof: #a " a 0 " 0 U is the midpoint of $ RT $ so the coordinates of U are %%, %% ! (0, 0). Thus S $U $ lies on

Given: Isosceles !RST; U is the midpoint of base $ RT $. Prove: S $U $⊥$ RT $

Example Prove that a segment from the vertex angle of an isosceles triangle to the midpoint of the base is perpendicular to the base. First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(#a, 0), and S(0, c). Then U(0, 0) is the midpoint of $ RT $.

Coordinate proofs can be used to prove theorems and to verify properties. Many coordinate proofs use the Distance Formula, Slope Formula, or Midpoint Theorem.

Triangles and Coordinate Proof

(continued)

____________ PERIOD _____

Study Guide and Intervention

Write Coordinate Proofs

4-7

NAME ______________________________________________ DATE

Answers (Lesson 4-7)

Glencoe Geometry

Lesson 4-7

©

Triangles and Coordinate Proof

Skills Practice

____________ PERIOD _____

Glencoe/McGraw-Hill

H(b, 0) x K(0, 0)

y

P (6b, 0) x

L(3b, c)

2. isosceles !KLP with base K $P $ 6b units long

A21

Q (4a, 0)

Q (?, ?) x

R(2a, b)

P (0, 0)

y

A(0, 2a)

C (0, 0) B(2a, 0) x

A(0, ?)

y

8.

5. Z (?, ?)

$

7 R &&b, c 2

#

Y (2b, 0) x

P (7b, 0) x

R(?, ?)

N (0, 0)

y

Z (b, c)

X(0, 0)

y

9.

6. y

T (0, b)

S (–a, 0)

y

N (3b, 0) x

U (a, 0) x

T (?, ?)

M (0, c)

O (0, 0)

M (?, ?)

D(5a, 0) x

N # 5–2a, b$

A(0, 0)

y

3. isosceles !AND with base A $D $ 5a long

©

Glencoe/McGraw-Hill

!M !⊥A !C !. of the slopes is %1, so B 221

Glencoe Geometry

y

C (2a, 0) x

M

Glencoe Geometry

B (0, 0)

A(0, 2a)

Answers

2a % 0 a%0 && " %1. The slope of B !M ! is && " 1. The product 0 % 2a a%0

M are #&&, &&$ or (a, a). The slope of A !C ! is

0 $ 2a 2a $ 0 2 2

The Midpoint Formula shows that the coordinates of

Proof:

Given: isosceles right !ABC with "ABC the right angle and M the midpoint of $ AC $ Prove: B $M $⊥$ AC $

10. Write a coordinate proof to prove that in an isosceles right triangle, the segment from the vertex of the right angle to the midpoint of the hypotenuse is perpendicular to the hypotenuse.

7.

4.

Find the missing coordinates of each triangle.

G(0, 0)

F (0, a)

y

1. right !FGH with legs a units and b units

Sample answers Position and label each triangle on the coordinate plane. are given.

4-7

NAME ______________________________________________ DATE

(Average)

Triangles and Coordinate Proof

Practice

4

L(3b, 0) x

P # 3–2b, c$

B(0, 0)

y

2. isosceles !BLP with base $ BL $ 3b units long

#

$

1 S &&b, c 6

R # 1–3b, 0$ x

S (?, ?)

J (0, 0)

y

5.

C (?, 0) x

C(3a, 0), E(0, c)

B (–3a, 0)

y

E (0, ?)

6.

P (2b, 0) x

x

M(0, c), N(%2b, 0)

N (?, 0)

M (0, ?)

y

G(0, 0) J (2a, 0)

D (0, 2a)

y

3. isosceles right !DGJ with hypotenuse D $J $ and legs 2a units long

Sample answers are given.

____________ PERIOD _____

©

M (–2, 3) S (0, 0)

y

K (6, 4)

x

Glencoe/McGraw-Hill

222

Glencoe Geometry

KM " %! (%2 %! 6)2 $! (3 % ! 4)2 " %! 64 $ 1 ! " %65 ! or ' 8.1 miles

8. Find the distance between the mall and Karina’s home.

Since the slope of S !M ! is the negative reciprocal of the slope of S !K !, S !M !⊥S !K !. Therefore, !SKM is right triangle.

Slope of SK " && or &&

4%0 2 6%0 3 3%0 3 Slope of SM " && or % && %2 % 0 2

Proof:

Given: !SKM Prove: !SKM is a right triangle.

7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall are at the vertices of a right triangle.

Karina lives 6 miles east and 4 miles north of her high school. After school she works part time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.

NEIGHBORHOODS For Exercises 7 and 8, use the following information.

4.

Find the missing coordinates of each triangle.

W # 1–4a, 0$ x

Y # 1–8a, b$

S(0, 0)

y

1. equilateral !SWY with 1 sides %% a long

Position and label each triangle on the coordinate plane.

4-7

NAME ______________________________________________ DATE

Answers (Lesson 4-7)

Lesson 4-7

©

Glencoe/McGraw-Hill

with "C as the vertex angle.

x

T(a, ?)

#

A22

D (?, ?)

y

F (?, ?)

x

R (–a, 0)

Sample answer: !RST is an isosceles right triangle. "RST is the right angle and is also the vertex angle. g. Find m"SRT and m"STR. 45; 45 h. Find m"OSR and m"OST. 45; 45

f. Combine your answers from parts c and e to describe !RST as completely as possible.

Sample answer: !RST is isosceles with "RST as the vertex angle.

e. What does your answer from part d tell you about !RST?

SR " %! 2a 2 or a%2 !; ST " %! 2a 2 or a%2 !; S !R !"S !T !

d. Find SR and ST. What does this tell you about S $R $ and S $T $?

Sample answer: !RST is a right triangle with "S as the right angle.

c. What does your answer from part b tell you about !RST ?

O (0, 0) T (a, 0) x

S (0, a)

y

D (0, 0), E (0, a), F (a, a)

E (?, a)

b. Find the product of the slopes of S $R $ and S $T $. What does this tell you about S $R $ and $ ST $ ? %1; S !R !⊥S !T !

$R $ and the slope of S $T $. 1; %1 a. Find the slope of S

2. Refer to the figure.

Glencoe/McGraw-Hill

223

Glencoe Geometry

Sample answer: Slope Formula: change in y over change in x ; Midpoint Formula: average of x-coordinates, average of y-coordinates

3. Many students find it easier to remember mathematical formulas if they can put them into words in a compact way. How can you use this approach to remember the slope and midpoint formulas easily?

Helping You Remember

©

$

b R (0, b), S (0, 0), T a, && 2

S (?, ?)

R (?, b)

y

1. Find the missing coordinates of each triangle. b.

From the coordinates of A, B, and C in the drawing in your textbook, what do you know about !ABC? Sample answer: !ABC is isosceles

Read the introduction to Lesson 4-7 at the top of page 222 in your textbook.

How can the coordinate plane be useful in proofs?

Triangles and Coordinate Proof

Reading the Lesson

a.

____________ PERIOD _____

Reading to Learn Mathematics

Pre-Activity

4-7

NAME ______________________________________________ DATE

Enrichment

5

8

5.

2.

13

40

6.

3.

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20; JKL, JKM, JKN, JKO, JLM, JLN, JLO, JMN, JMO, JNO, KLM, KLN, KLO, KMN, KMO, KNO, LMN, LMO, LNO, MNO

O

J

4; ABC, ABD, ACD, BCD

A

B

Glencoe/McGraw-Hill

8.

7.

224

9.

8.

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Glencoe Geometry

35; PQR, PQS, PQT, PQU, PQV, PRS, PRT, PRU, PRV, PST, PSU, PSV, PTU, PTV, PUV, QRS, QRT, QRU, QRV, QST, QSU, QSV, QTU, QTV, QUV, RST, RSU, RSV, RTU, RTV, RUV, STU, STV, SUV, TUV

P

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10; EFG, EFH, EFI, EGH, EHI, FGH, FGI, FHI, EGI, GHI

I

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27

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How many triangles can you form by joining points on each circle? List the vertices of each triangle.

4.

1.

How many triangles are there in each figure?

Each puzzle below contains many triangles. Count them carefully. Some triangles overlap other triangles.

How Many Triangles?

4-7

NAME ______________________________________________ DATE

Answers (Lesson 4-7)

Glencoe Geometry

Lesson 4-7