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Understanding Inscribed Angles and Arc Measures in Circles

This resource explores the concepts of inscribed angles and the relationships between angles and intercepted arcs within circles. Examples illustrate how to determine angle measures, such as in semicircles, and how to apply Theorem 10.7 for calculating angle relationships. The document highlights the supplementary nature of opposite angles in inscribed quadrilaterals. Practical exercises reinforce these concepts, providing a comprehensive overview for students learning about circle geometry.

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Understanding Inscribed Angles and Arc Measures in Circles

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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 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 Find the measure of an intercepted arc SOLUTION

  3. a. m G = mHF = (90o) = 45o 1 1 2 2 EXAMPLE 3 Find the measure of the red arc or angle. 3. SOLUTION

  4. mTV = 2m U = 2 38o = 76o. b. EXAMPLE 4 Find the measure of the red arc or angle. 2. SOLUTION

  5. a. PQRS is inscribed in a circle, so opposite angles are supplementary. a. mQ + m S = 180o m P + m R = 180o EXAMPLE 5 Find the value of each variable. SOLUTION 75o + yo = 180o 80o + xo = 180o y = 105 x = 100

  6. 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 10.10 Find the value of each variable. SOLUTION 4bo + 2bo = 180o 2ao + 2ao = 180o 6b = 180 4a = 180 b = 30 a = 45

  7. Assignment… Using your textbook: Pg. 207 #1-18

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