10.5 Angles Related To A Circle

1. 10.5 Angles Related To A Circle After studying this section, you will be able to determine the measures central angles, inscribed and tangent-chord angles, chord-chord angles, secant-secant, secant-tangent and tangent-tangent angles.

2. Angles with Vertices at the Center of a Circle An angle with its vertex at the center of a circle is a central angle, already defined to be equal in measure to its intercepted arc (10.3). A 50 B 50 O

3. Angles with Vertices on a Circle A C B O O E D Definition An inscribed angle is an angle whose vertex is on a circle and whose sides are determined by two chords. Definition A tangent-chord angle is an angle whose vertex is on the circle and whose sides are determined by a tangent and a chord that intersect at the tangent’s point of contact.

4. Theorem The measure of an inscribed angle or a tangent-chord angle (vertex on the circle) is one-half the measure of its intercepted arc. Example 1 The measure of arc AC is 112. Find the measure of angle B. A C B

5. Example 2 FE is tangent at E and the measure of arc DE = 80. Find the measure of angle DEF E F D

6. Angles with Vertices Inside, but not at the center of a circle Definition A chord-chord angle is an angle formed by two chords that intersect inside a circle but not at the center. Theorem The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle. D C P O F E

7. Angles with Vertices Outside a Circle Definition A secant-secant angle is an angle whose vertex is outside a circle and whose sides are determined by two secants. B A C D E

8. Definition A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent. Y Z X W

9. Definition A tangent-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by two tangents. H K J

10. Theorem The measure of a secant-secant angle, a secant-tangent angle or a tangent-tangent angle (vertex outside a circle) is one-half the difference of the measures of the intercepted arcs. Example 3 Find the measure of angle E B 100 A C 20 D E

11. Example 4 Find the measure of angle W Y 100 Z 60 X W H Example 5 Find the measure of angle J 100 K J

12. Example 6 Find y D 29° A y C 47° B

13. Example 7 Find the measure of arcs AB and CD Hint: Let arc AB = x and arc CD = y A D E F 24° 65° C B

14. Summary Using pictures write the rules to find each of the different types of angles. Homework: worksheet