1. Other Angles in Circles 10.4 California State Standards 7: Prove/use theorems involving circles. 21: Prove/solve relationships with circles.

2. theorem Tangent-Chord Angles The measure of an angle formed by a tangent and a chord at a point on a circle equals ½ the measure of the intercepted arc. C B A 1 2

3. A example B C 204 o 102 o Find the measure of the arc or angle.

4. theorem Chord-Chord Angles The measure of an angle formed by two intersecting chords equals ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle. C Q P R S 1

5. B A C xoxo 120 o D 40 o example Find the value of x

6. theorem Tangent and Secant Angles The measure of an angle formed from tangents and/or secants intersecting in the exterior of a circle S Q P 1 R T

7. theorem Tangent and Secant Angles The measure of an angle formed from tangents and/or secants intersecting in the exterior of a circle S Q P 1 R T

8. theorem Tangent and Secant Angles The measure of an angle formed from tangents and/or secants intersecting in the exterior of a circle S Q 1 R T P

9. P R T Q S 1 Tangent and Secant Angles The measure of an angle formed from tangents and/or secants intersecting in the exterior of a circle equals ½ the difference of the measures of the intercepted arcs. theorem S R T Q P 1 R Q S 1 T P

10. example S Q P R x 80 o Find the value of x

11. Measures of Angles & Arcs Summary An angle with its vertex on the circle measures ½ the intercepted arc. An angle with its vertex inside the circle measures ½ the sum of its intercepted arc and its vertical’s intercepted arc. An angle with its vertex outside the circle measures ½ the difference of the two intercepted arcs.

12. example B A C 3xo3xo (4 x + 50) o Find the measure of the arc or angle

13. examples xoxo 170 o 30 o 78 o yoyo Find the value of x and y 82 o

14. proof R Q N K J P Statement Reason