95 Inscribed Angles

1
Chapter 10
Angle Formulas
2
Central Angle
Definition
An angle whose vertex lies on the center of the
circle.
NOT A Central Angle (of a circle)
Central Angle (of a circle)
Central Angle (of a circle)
3
Central Angle Theorem

• The measure of a center angle is equal to the
  measure of the intercepted arc.

Center Angle
Intercepted Arc
Example
Give is the diameter, find the value of x
and y and z in the figure.
4
Example Find the measure of each arc.
4x 3x (3x 10) 2x (2x-14) 360
14x 4 360
14x 364
x 26
4x 4(26) 104 3x 3(26) 78 3x 10 3(26)
10 88 2x 2(26) 52 2x 14 2(26) 14
38
5
Inscribed Angle
Inscribed Angle An angle whose vertex lies on a
circle and whose sides are chords of the circle
(or one side tangent to the circle).
Examples
3
1
2
4
No!
Yes!
No!
Yes!
6
Intercepted Arc
Intercepted Arc An angle intercepts an arc if
and only if each of the following conditions
holds
1. The endpoints of the arc lie on the angle. 2.
All points of the arc, except the endpoints, are
in the interior of the angle. 3. Each side of the
angle contains an endpoint of the arc.
7
Inscribed Angle Theorem
The measure of an inscribed angle is equal to ½
the measure of the intercepted arc.
Y
Inscribed Angle
110?
55?
Z
Intercepted Arc
An angle formed by a chord and a tangent can be
considered an inscribed angle.
8
Examples Find the value of x and y in the fig.
Chapter 10
9
An angle inscribed in a semicircle is a right
angle.
P
180?
90?
S
R
10
Interior Angle Theorem
Angles that are formed by two intersecting chords.
Definition
are interior angles.
2
E
Interior Angle Theorem
The measure of the angle formed by the two
intersecting chords is equal to ½ the sum of the
measures of the intercepted arcs.
11
Example Interior Angle Theorem
91?
A
C
x
y
B
D
85?
12
Exterior Angles
An angle formed by two secants, two tangents, or
a secant and a tangent drawn from a point outside
the circle.
Two secants
2 tangents
A secant and a tangent
13
Exterior Angle Theorem
The measure of the angle formed is equal to ½ the
difference of the intercepted arcs.
14
Example Exterior Angle Theorem
15
30
25
100
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Inscribed Quadrilaterals
If a quadrilateral is inscribed in a circle, then
the opposite angles are supplementary.
m?DAB m?DCB 180 ? m?ADC m?ABC 180 ?