Inscribed Angles

1. Inscribed Angles Geometry CP2 (Holt 12-4) K. Santos

2. Inscribed angle Inscribed angle—an angle with its vertex on the circle A C B < C is and inscribed angle is the intercepted arc

3. Inscribed Angle Theorem 12-4-1 Inscribed angle = half the measure of its intercepted arc. X Y Z m < Y = ½

4. Example—Inscribed Angle Find the values of a and b. 32 b a a = ½ of a semicircle a = ½ (180) = 90 The inscribed angle has an arc 2(32) = 64 b= 180 – 64 b = 116

5. Example—Inscribed angles P a Find the values of a and b. T 30 60 S Q b R m < PQT = ½ m 60 = ½ a 120= a m < PRS = ½ m b = ½ (m +m) b = ½ (120 + 30) b = ½ (150) b = 75

6. Corollary to the Inscribed Angle Theorem 12-4-2 Two inscribed angles that intercept the same arc are congruent. A D B C <A and < D have the same intercepted arcs so, <A <D

7. Theorem 12-4-3 An angle inscribed in a semicircle is a right angle. M N P O < M has an intercepted arc of this arc is a semicircle So, m < M = 90 (1/2 of 180)

8. Theorem 12-4-4 Quadrilateral is inscribed in a circle = opposite angles aresupplementary. A B C D < A and <C are opposite angles, so they are supplementary <B and <D are opposite angles, so they are supplementary

9. Example—Inscribed quadrilateral m < A = 70 and m < B = 120. Find m < C and m < D A B C D < A and < C are supplementary, 180 – 70 =110 so m<C = 110 < B and < D are supplementary 180 – 120 = 60 so m <D = 60