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Angles Related to a Circle. Lesson 10.5. Inscribed Angle: an angle whose vertex is on a circle and whose sides are determined by two chords. - PowerPoint PPT Presentation
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Angles Related to a Angles Related to a CircleCircleLesson 10.5
Angles with Vertices on a CircleAngles 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 (88 + 27)
x = 57.5º
½ a = 65
a = 130
½ (21 + y) = 72
21 + y = 144
y = 123º
Find y.1. Find mBEC.
2. mBEC = ½ (29 + 47)
3. mBEC = 38º
4. y = 180 – mBEC
5. y = 180 – 38 = 142º
Part 2 of Section 10.5…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.
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) = 32
125 – z = 64
z = 61
1. First find the measure of arc EA.
2. m of arc AEB = 180 so arc EA = 180 – (104 + 20) = 56
3. .
4. mC = ½ (56 – 20)
5. mC = 18
½ (x + y) = 65 and ½ (x – y ) = 24
x + y = 130 and x – y = 48
x + y = 130x – y = 48 2x = 178 x = 89
89 + y = 130 y = 41
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