EXAMPLE 1 Use inscribed angles

a. m T mQRb.

Find the indicated measure in P.

SOLUTION

12

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

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

mTQ = 2m R = 2 50o = 100o. Because TQR is a semicircle,b.

EXAMPLE 2 Find the measure of an intercepted arc

Find mRS and m STR. What do you notice about STR and RUS?

SOLUTION

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

Also, m STR = mRS = (62o) = 31o. So, STR RUS.12

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EXAMPLE 3 Standardized Test Practice

SOLUTION

Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem 6.10. Also, KJLand KML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles.

GUIDED PRACTICE for Examples 1, 2 and 3

Find the measure of the red arc or angle.

1.

SOLUTION

m G = mHF = (90o) = 45o12

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a.

GUIDED PRACTICE for Examples 1, 2 and 3

Find the measure of the red arc or angle.

2.

SOLUTION

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

GUIDED PRACTICE for Examples 1, 2 and 3

Find the measure of the red arc or angle.

3.

SOLUTION

ZYN ZXN

ZXN 72°

Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem 6.10. Also, KJL and KML intercept the same arc, so they must also be congruent.

ZXN

EXAMPLE 4 Use a circumscribed circle

PhotographyYour 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

SOLUTION

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.

GUIDED PRACTICE for Example 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.

4.

Make the diameter of your circle the diagonal of the rectangular base.

SOLUTION

EXAMPLE 5 Use Theorem 6.12

Find the value of each variable.

a.

SOLUTION

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

a.

m P + m R = 180o

75o + yo = 180o

y = 105

m Q + m S = 180o

80o + xo = 180o

x = 100

EXAMPLE 5 Use Theorem 6.12

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

m J + m L = 180o

2ao + 2ao = 180o

a = 45

m K + m M = 180o

4bo + 2bo = 180o

b = 30

4a = 180 6b = 180

Find the value of each variable.

b.

SOLUTION

GUIDED PRACTICE for Example 5

5.

Find the value of each variable.

SOLUTION

y = 112 x = 98

GUIDED PRACTICE for Example 5

6.

Find the value of each variable.

SOLUTION

c = 62 x = 10