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) ©
Glencoe/McGraw-Hill
vii
Glencoe Geometry
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
©
Glencoe/McGraw-Hill
viii
Glencoe Geometry
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) ©
Glencoe/McGraw-Hill
ix
Glencoe Geometry
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)
©
Glencoe/McGraw-Hill
x
Glencoe Geometry
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.
©
Glencoe/McGraw-Hill
184
Glencoe Geometry
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)
©
Glencoe/McGraw-Hill
185
Glencoe Geometry
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?
©
Glencoe/McGraw-Hill
186
Glencoe Geometry
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?
©
Glencoe/McGraw-Hill
187
Glencoe Geometry
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?
Glencoe/McGraw-Hill
188
Glencoe Geometry
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.
©
Glencoe/McGraw-Hill
192
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.
©
Glencoe/McGraw-Hill
193
Glencoe Geometry
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.
©
Glencoe/McGraw-Hill
194
Glencoe Geometry
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.
©
Glencoe/McGraw-Hill
198
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?
©
Glencoe/McGraw-Hill
199
Glencoe Geometry
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)
©
Glencoe/McGraw-Hill
&
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.
©
T
Glencoe/McGraw-Hill
201
Glencoe Geometry
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
©
Glencoe/McGraw-Hill
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.
©
Glencoe/McGraw-Hill
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$
©
Glencoe/McGraw-Hill
204
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.
©
Glencoe/McGraw-Hill
205
Glencoe Geometry
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?
©
Glencoe/McGraw-Hill
206
Glencoe Geometry
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
©
Glencoe/McGraw-Hill
207
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 "
©
Glencoe/McGraw-Hill
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
©
Glencoe/McGraw-Hill
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.
©
Glencoe/McGraw-Hill
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?
©
Glencoe/McGraw-Hill
211
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.
©
Glencoe/McGraw-Hill
212
Glencoe Geometry
NAME ______________________________________________ DATE
____________ 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.
©
Glencoe/McGraw-Hill
216
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.
©
Glencoe/McGraw-Hill
217
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
©
Y
Glencoe/McGraw-Hill
X
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)
©
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)
©
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.
©
Glencoe/McGraw-Hill
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?
©
Glencoe/McGraw-Hill
223
Glencoe Geometry
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%
©
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
©
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. ©
Glencoe/McGraw-Hill
237
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.
©
Glencoe/McGraw-Hill
238
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%
©
Glencoe/McGraw-Hill
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.
©
Glencoe/McGraw-Hill
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
©
Glencoe/McGraw-Hill
241
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
Glencoe/McGraw-Hill
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
©
Glencoe/McGraw-Hill
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
©
Glencoe/McGraw-Hill
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
©
____________ PERIOD _____
Reading to Learn Mathematics
Pre-Activity
4-6
NAME ______________________________________________ DATE
Enrichment
©
F
B
E
C
D
Y
Glencoe/McGraw-Hill
Z
U
X
V
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
J
N
M
Glencoe Geometry
L
4. Given: m"N ! 120, J N#M $$ $N $, !JNM # !KLM. Find m"JKM. 15
D
E
A
2. Given: AC ! AD, and A $B $#B $D $, m"DAC ! 44 and C $E $ bisects " ACD. Find m"DEC. 78
K
____________ PERIOD _____
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)
©
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.
©
D
C
N
M
K
L
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.
H
G
F
V
U
T
R
S
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
Q
10; EFG, EFH, EFI, EGH, EHI, FGH, FGI, FHI, EGI, GHI
I
E
27
35
____________ PERIOD _____
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
