1. Angles Related to a Circle
Lesson 12.6

2. 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: 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

4. 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.

5. ½ a = 65
a = 130
x = ½ ( )
x = 57.5º

6. ½ (21 + y) = 72
21 + y = 144
y = 123º

7. Find y. Find mBEC. mBEC = ½ (29 + 47) mBEC = 38º y = 180 – mBEC

8. 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.

9. 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.

10. 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: 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.

11. y = ½ (57 – 31)
y = ½(26)
y = 13
½ (125 – z) = 32
125 – z = 64
z = 61

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

13. 360 – 134 = 226
(226 – 134)/2 = ?
46 = ?

14. ½[(360 – x) – x] = 70
(360 – x) – x = 140
360 – 2x = 140
360 – 140 = 2x
110 = x
Check: (250 – 110)/2 = 70

15. 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