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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.

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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 on a Circle
slide4

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.

slide5

½ a = 65

a = 130

x = ½ (88 + 27)

x = 57.5º

slide6

½ (21 + y) = 72

21 + y = 144

y = 123º

slide7

Find y.

Find mBEC.

mBEC = ½ (29 + 47)

mBEC = 38º

y = 180 – mBEC

y = 180 – 38 = 142º

slide9

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.

slide10

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.

slide11

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.

slide12

y = ½ (57 – 31)

y = ½(26)

y = 13

½ (125 – z) = 32

125 – z = 64

z = 61

slide13

First find the measure of arc EA.

m of arc AEB = 180 so arc EA = 180 – (104 + 20) = 56

.

mC = ½ (56 – 20)

mC = 18

slide14

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 = 89

89 + y = 130

y = 41