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MM2G3b Understand and use properties of central, inscribed, and related angles

Standard. MM2G3b Understand and use properties of central, inscribed, and related angles. Theorem 6.9. The measure of an inscribed angle is one half the measure of its intercepted arc. Find the indicated measure in P. a. m T. b. mQR. a. M T = mRS = (48 o ) = 24 o.

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MM2G3b Understand and use properties of central, inscribed, and related angles

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  1. Standard MM2G3b Understand and use properties of central, inscribed, and related angles

  2. Theorem 6.9 The measure of an inscribed angle is one half the measure of its intercepted arc

  3. 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

  4. Theorem 6.10 If two inscribed angles of a circle intercept the same arc, then the angles are congruent

  5. 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 SOLUTION

  6. 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 SOLUTION

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

  8. mTV = 2m U = 2 38o = 76o. b. GUIDED PRACTICE Find the measure of the red arc or angle. 2. SOLUTION

  9. 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° GUIDED PRACTICE Find the measure of the red arc or angle. 3. SOLUTION

  10. Theorem 6.11 If a right triangle is inscribed in a circle (all vertices lie on the circle) then the hypotenuse is the diameter of the circle.

  11. 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 Photography

  12. 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 SOLUTION

  13. Theorem 6.12 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

  14. 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

  15. b. JKLMis inscribed in a circle, so opposite angles are supplementary. b. mK + m M = 180o m J + m L = 180o EXAMPLE 5 Find the value of each variable. SOLUTION 4bo + 2bo = 180o 2ao + 2ao = 180o 6b = 180 4a = 180 b = 30 a = 45

  16. GUIDED PRACTICE Find the value of each variable. 4. SOLUTION y = 112 x = 98

  17. GUIDED PRACTICE Find the value of each variable. 5. SOLUTION c = 62 x = 10

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