11.5 Inscribed Angles and Polygons

Page 1 of 6

11.5 Goal Use properties of inscribed angles.

Inscribed Angles and Polygons An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

inscribed angle

The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

Key Words • inscribed angle • intercepted arc

intercepted arc

Activity 11.5 shows the relationship between an inscribed angle and its intercepted arc.

• inscribed • circumscribed

THEOREM 11.7

Measure of an Inscribed Angle Words

A

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

Symbols

EXAMPLE

C D

B

1 maADB  mAB s 2

Find Measures of Inscribed Angles and Arcs

1

Find the measure of the inscribed angle or the intercepted arc. a.

b.

N M

W

100ⴗ Z P

105ⴗ

X

Y

Solution

IStudent Help ICLASSZONE.COM

1 2

The measure of an inscribed angle is half the measure of its intercepted arc.

 (100ⴗ)

1 2

Substitute 100ⴗ for mNP s.

 50

Simplify.

a. maNMP  mNP s

MORE EXAMPLES More examples at classzone.com

1 2

The measure of an inscribed angle is half the measure of its intercepted arc.

105ⴗ  mZWX t

1 2

Substitute 105ⴗ for maZYX.

210  mZWX t

Multiply each side by 2.

b. maZYX  mZWX t

614

Chapter 11

Circles

Page 2 of 6

Find Measures of Inscribed Angles and Arcs Find the measure of the inscribed angle or the intercepted arc. 1.

2.

B

3.

D

90ⴗ

N P

160ⴗ

C

120ⴗ

A

E

F

M

K

Inscribed and Circumscribed If all the vertices of a polygon lie on

a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle. circumscribed circle inscribed triangle

inscribed quadrilateral

THEOREM 11.8 Words

If a triangle inscribed in a circle is a right triangle, then the hypotenuse is a diameter of the circle. If a side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle.

EXAMPLE

2

B C

Find Angle Measures

Find the values of x and y.

C xⴗ

Solution

LOOK BACK

&* is a Because T ABC is inscribed in a circle and AB diameter, it follows from Theorem 11.8 that T ABC &*. is a right triangle with hypotenuse AB

To review the Corollary of the Triangle Sum Theorem, see p. 180.

Therefore, x  90. Because aA and aB are acute angles of a right triangle, y  90  50  40.

Student Help

A

11.5

A

yⴗ

50ⴗ D

Inscribed Angles and Polygons

B

615

Page 3 of 6

Find Angle Measures Find the values of x and y in 䉺C. 4.

5.

xⴗ

6.

xⴗ

yⴗ

35ⴗ C

C

yⴗ

C xⴗ

Visualize It!

THEOREM 11.9

aD and aF are opposite angles. aE and aG are opposite angles.

Words

F

E

If a quadrilateral can be inscribed in a circle, then its opposite angles are supplementary.

F

E C

If the opposite angles of a quadrilateral are supplementary, then the quadrilateral can be inscribed in a circle.

C D

xⴗ

60ⴗ yⴗ

D

G

G

3

EXAMPLE

Find Angle Measures

Find the values of y and z.

R zⴗ

Solution

yⴗ

Because RSTU is inscribed in a circle, by Theorem 11.9 opposite angles must be supplementary.

S 120ⴗ 80ⴗ T

U

aS and aU are opposite angles.

aR and aT are opposite angles.

maS  maU  180

maR  maT  180

120ⴗ  yⴗ  180

zⴗ  80ⴗ  180

y  60

z  100

Find Angle Measures Find the values of x and y in 䉺C. 7.

8.

xⴗ C 100ⴗ

Chapter 11

Circles

yⴗ

yⴗ

yⴗ C

50ⴗ C

xⴗ 80ⴗ

95ⴗ

616

9.

xⴗ

Page 4 of 6

11.5 Exercises Guided Practice Vocabulary Check

In Exercises 1 and 2, use the diagram at the right.

B

1. Name the inscribed angles.

C

2. Identify the two pairs of opposite angles

in the inscribed quadrilateral.

Skill Check

D

A

Find the measure of the blue intercepted arc. 3.

4.

L

K

5.

K

J 105ⴗ

M

20ⴗ

K

L L

J

J

M

Find the value of each variable. 6.

7.

8.

75ⴗ

yⴗ

xⴗ

85ⴗ

80ⴗ

230ⴗ yⴗ

xⴗ zⴗ

Practice and Applications Extra Practice See p. 696.

Angle Measures Find the measure of the inscribed angle. A

9.

10.

110ⴗ B

Example 1: Exs. 9–27 Example 2: Exs. 28–31 Example 3: Exs. 32–38

180ⴗ B

C

12.

218ⴗ

13.

L

A

B

C

Homework Help

11.

A

M

14.

P P 134ⴗ

68ⴗ R

N

11.5

C

238ⴗ U T

S

Inscribed Angles and Polygons

617

Page 5 of 6

Arc Measures Find the measure of the blue intercepted arc. 15.

16.

17.

B

B

A 32ⴗ

114ⴗ A 78ⴗ C

18.

A

B

C

C

19.

S T 120ⴗ

20.

W

P

R

50ⴗ U

P

Z 103ⴗ Y

X

N

Arc and Angle Measures In Exercises 21–26, use the diagram below to find the intercepted arc or inscribed angle.

Student Help VISUAL STRATEGY

In Exs. 21–26, copy the diagram and add information to it as you solve the exercises, as shown on p. 588.

21. mBE r

22. maBDE

23. maAED

24. mAD r

25. maABD

26. mDE r

A D E

27. Are T ABC and T DEC similar?

C

47ⴗ

80ⴗ

100ⴗ B

Explain your reasoning. Inscribed Right Triangles Find the value of each variable. Explain your reasoning. 28.

A

29.

30ⴗ D yⴗ C xⴗ B

M

L xⴗ yⴗ

40ⴗ

30.

P

xⴗ

yⴗ

S

58ⴗ

P

R

K

31. Carpenter’s Square A carpenter’s square is an L-shaped tool used

to draw right angles. Suppose you are making a toy truck. To make the wheels you trace a circle on a piece of wood. How could you use a carpenter’s square to find the center of the circle?

IStudent Help ICLASSZONE.COM

HOMEWORK HELP Extra help with problem solving in Ex. 31 is at classzone.com

618

Chapter 11

Circles

Page 6 of 6

Inscribed Quadrilaterals Find the values of x and y. 32.

33.

A

114ⴗ

yⴗ

A xⴗ

92ⴗ B yⴗ

A

B

xⴗ

D

34.

115ⴗ yⴗ

D

xⴗ

102ⴗ 100ⴗ C D

C

B

C

You be the Judge

Can the quadrilateral always be inscribed in a circle? Explain your answer.

Standardized Test Practice

35. square

36. isosceles trapezoid

37. rhombus

38. rectangle

39. Multiple Choice In the diagram at the right,

if aACB is a central angle and maACB  80, what is maADB? A 

20

B 

40

C 

80

D 

160

B D

A

40. Multiple Choice In the diagram at the right,

100ⴗ

what are the values of x and y?

Mixed Review

C

F 

x  80, y  95

G 

x  85, y  100

H 

x  95, y  80

J 

x  95, y  85

85ⴗ

xⴗ

yⴗ

Multiplying Radicals Multiply the radicals. Then simplify if possible. (Lesson 10.1) 41. 兹5 苶 p 兹7 苶

(

42. 兹2 苶 p 兹2 苶

)

(

44. 8兹2 苶 2

43. 兹6 苶 p 兹1 苶4 苶

)

45. 3兹3 苶 2

46. 2兹5 苶 p 兹1 苶0 苶

Solving Right Triangles Solve the right triangle. Round decimals to the nearest tenth. (Lesson 10.6) 47.

B

48. J

49. R

11

8 A

Algebra Skills

44ⴗ

C

P 50ⴗ

35ⴗ

K

5 P

L

Evaluating Expressions Evaluate the expression when x ⴝ 2. (Skills Review, p. 670) 50. 3x  5

51. 8x  7

52. x2  9

53. (x  4)(x  4)

54. x2  3x  2

55. x3  x2

11.5

Inscribed Angles and Polygons

619