Inscribed Angles

1. Inscribed Angles Geometry H2(Holt 12-4) K. Santos

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

3. Inscribed Angle Theorem 12-4-1 The measure of an inscribed angle is 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

5. Example—Inscribed angles P a Find the values of a and b. T 30 60 S Q b R

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 If a quadrilateral is inscribed in a circle, then its 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

10. Example—Inscribed Quadrilateral E Given m = 70, m = 80, and m 90. Find m < G and m <D. D F G