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Find the indicated measure in P. a. m T. b. mQR. a. M T = mRS = (48 o ) = 24 o. mTQ = 2 m R = 2 50 o = 100 o . Because TQR is a semicircle,. b. mQR = 180 o mTQ = 180 o 100 o = 80 o . So, mQR = 80 o. –. –. 1. 1. 2. 2. EXAMPLE 1.

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EXAMPLE 1

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Example 1

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

Use inscribed angles

SOLUTION


Example 1

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

From Theorem 6.9,you know thatmRS = 2m RUS= 2 (31o) = 62o.

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

1

1

2

2

EXAMPLE 2

Find the measure of an intercepted arc

SOLUTION


Example 1

Notice thatJKM andJLM intercept the same arc, and soJKM JLM by Theorem 6.10. Also, KJLandKML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles.

EXAMPLE 3

Standardized Test Practice

SOLUTION


Example 1

a.

m G = mHF = (90o) = 45o

1

1

2

2

for Examples 1, 2 and 3

GUIDED PRACTICE

Find the measure of the red arc or angle.

1.

SOLUTION


Example 1

mTV = 2m U = 2 38o = 76o.

b.

for Examples 1, 2 and 3

GUIDED PRACTICE

Find the measure of the red arc or angle.

2.

SOLUTION


Example 1

Notice thatZYN andZXN intercept the same arc, and soZYN byTheorem 6.10. Also, KJL and KML intercept the same arc, so they must also be congruent.

ZXN

ZYN

ZXN

ZXN

72°

for Examples 1, 2 and 3

GUIDED PRACTICE

Find the measure of the red arc or angle.

3.

SOLUTION


Example 1

Your camera has a 90o 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?

EXAMPLE 4

Use a circumscribed circle

Photography


Example 1

From Theorem 6.11, 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.

EXAMPLE 4

Use a circumscribed circle

SOLUTION


Example 1

for Example 4

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.

SOLUTION

Make the diameter of your circle the diagonal of the

rectangular base.


Example 1

a.

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

a.

mQ + m S = 180o

m P + m R = 180o

EXAMPLE 5

Use Theorem 6.12

Find the value of each variable.

SOLUTION

75o + yo = 180o

80o + xo = 180o

y = 105

x = 100


Example 1

b.

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

b.

mK + m M = 180o

m J + m L = 180o

EXAMPLE 5

Use Theorem 6.12

Find the value of each variable.

SOLUTION

4bo + 2bo = 180o

2ao + 2ao = 180o

6b = 180

4a = 180

b = 30

a = 45


Example 1

for Example 5

GUIDED PRACTICE

Find the value of each variable.

5.

SOLUTION

y = 112

x = 98


Example 1

for Example 5

GUIDED PRACTICE

Find the value of each variable.

6.

SOLUTION

c = 62

x = 10


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