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Angles Related to a Circle
NOTES 10.5 Angles Related to a Circle
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INSCRIBED Angles 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.
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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.
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½ a = 65 a = 130 x = ½ ( ) x = 57.5º
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½ (21 + y) = 72 21 + y = 144 y = 123º
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Find y. Find mBEC. mBEC = ½ (29 + 47) mBEC = 38º y = 180 – mBEC
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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.
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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.
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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.
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y = ½ (57 – 31) y = ½(26) y = 13 ½ (125 – z) = 32 125 – z = 64 z = 61
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First find the measure of arc EA.
m of arc AEB = 180 so arc EA = 180 – ( ) = 56 . mC = ½ (56 – 20) mC = 18
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½ (x + y) = 65 and ½ (x – y ) = 24 x + y = and x – y = 48 x + y = 130 x – y = 48 2x = 178 x = 89 89 + y = 130 y = 41
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