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Use Inscribed Angles and Polygons. Warm Up. Lesson Presentation. Lesson Quiz. 25. ANSWER. 60º; 120º. ANSWER. Warm-Up. 1. The measure of the interior angles of a quadrilateral are 80º, 100º, 55º, and 5 x º . Find the value of x.

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Warm Up


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Use Inscribed Angles and Polygons

Warm Up

Lesson Presentation

Lesson Quiz

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25

ANSWER

60º; 120º

ANSWER

Warm-Up

1.The measure of the interior angles of a quadrilateralare 80º, 100º, 55º, and 5xº. Find the value of x.

2.Two supplementary angles have measures 6xº and12xº. Find each angle measure.

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3.Solve 3x = ( 4x + 12).

1 2

1 2

6

ANSWER

4. Solve 80 = ( 360 – 2x).

100

ANSWER

Warm-Up

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Find the indicated measure inP.

a.

mT

b.

mQR

a.

M T = mRS = (48o) = 24o

mTQ = 2m R = 2 50o = 100o.BecauseTQR is a semicircle,

b.

mQR = 180o mTQ = 180o 100o = 80o. So,mQR = 80o.

1

1

2

2

Example 1

SOLUTION

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Find mRSand mSTR. What do you notice about STRand RUS?

From Theorem 10.7,you know thatmRS = 2m RUS= 2 (31o) = 62o.

Also, m STR = mRS = (62o) = 31o. So,STR RUS.

1

1

2

2

Example 2

SOLUTION

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Notice thatJKM andJLM intercept the same arc, and soJKM JLM by Theorem 10.8. Also, KJL andKML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles.

So, by Theorem 10.8, the correct answer is C.

Example 3

SOLUTION

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1.

ANSWER

45°

Guided Practice

Find the measure of the red arc or angle.

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2.

ANSWER

76°

Guided Practice

Find the measure of the red arc or angle.

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3.

ANSWER

72°

Guided Practice

Find the measure of the red arc or angle.

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Example 4

Photography

Your camera has a 90° field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way?

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Example 4

SOLUTION

From Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. So, draw the circle that has the front of the statue as a diameter. The statue fits perfectly within your camera’s 90o field of vision from any point on the semicircle in front of the statue.

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ANSWER

Make the diameter of your circle the diagonal of the

rectangular base.

Guided Practice

4. WHAT IF? In Example 4,explain how to find locations if you want to frame the front and left side of the statue in your picture.

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a.

PQRS is inscribed in a circle, so opposite angles are supplementary.

a.

mQ + m S = 180°

m P + m R = 180°

Example 5

Find the value of each variable.

SOLUTION

75° + y° = 180°

80° + x° = 180°

y = 105

x = 100

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b.

JKLMis inscribed in a circle, so opposite angles are supplementary.

b.

mK + m M = 180°

m J + m L = 180°

Example 5

Find the value of each variable.

SOLUTION

2a° + 2a° = 180°

4b° + 2b° = 180°

4a = 180

6b = 180

a = 45

b = 30

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5.

ANSWER

x = 98, y = 112

Guided Practice

Find the value of each variable.

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6.

ANSWER

c = 62, x = 10

Guided Practice

Find the value of each variable.

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1.

ANSWER

ANSWER

38

56

Lesson Quiz

Find the value of x .

2.

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ANSWER

44

Lesson Quiz

Find the value of x .

3.

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ANSWER

x = 54; y = 20

Lesson Quiz

Find the value of each variable.

4.

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ANSWER

x = 5; y = 10

Lesson Quiz

Find the value of each variable.

5.