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EXAMPLE 1

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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  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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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.

  10. 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

  11. 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

  12. for Example 5 GUIDED PRACTICE Find the value of each variable. 5. SOLUTION y = 112 x = 98

  13. for Example 5 GUIDED PRACTICE Find the value of each variable. 6. SOLUTION c = 62 x = 10

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