Angles Related to a Circle Lesson 10.5

Angles with Vertices on a Circle Inscribed Angle: an angle whose vertex is on a circle and whose sides are determined by two chords. Tangent-Chord Angle: Angle whose vertex is on a circle whose sides are determined by a tangent and a chord that intersects at the tangent’s point of contact. Theorem 86: The measure of an inscribed angle or a tangent-chord angle (vertex on circle) is ½ the measure of its intercepted arc.

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 87: 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.

x = ½ ( ) x = 57.5º ½ a = 65 a = 130

½ (21 + y) = y = 144 y = 123 º

Find y. 1. Find m  BEC. 2. m  BEC = ½ ( ) 3. m  BEC = 38º 4. y = 180 – m  BEC 5. y = 180 – 38 = 142º

Part 2 of Section 10.5…

Angles with Vertices Outside a Circle Three types of angles… 1. A secant-secant angle is an angle whose vertex is outside a circle and whose sides are determined by two secants.

Angles with Vertices Outside a Circle 2. A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.

Angles with Vertices Outside a Circle 3. A tangent-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by two tangents. Theorem 88: The measure of a secant-secant angle, a secant-tangent angle, or a tangent- tangent angle (vertex outside a circle) is ½ the difference of the measures of the intercepted arcs.

y = ½ (57 – 31) y = ½(26) y = 13 ½ (125 – z) = – z = 64 z = 61

1. First find the measure of arc EA. 2. m of arc AEB = 180 so arc EA = 180 – ( ) = m  C = ½ (56 – 20) 5. m  C = 18

½ (x + y) = 65 and ½ (x – y ) = 24 x + y = 130 and x – y = 48 x + y = 130 x – y = 48 2x = 178 x = y = 130 y = 41