10.4 Inscribed Angles

5/7/2010

Using Inscribed Angles• An inscribed

angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

• Intercepted arc is an arc that lies in the interior of an inscribed angle and has endpoints on the angle.

intercepted arc

inscribed angle

To find the measure of an arc use the central angle.

Central angle

115˚

115˚

Theorem 10.7: Measure of an Inscribed Angle

• If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc.

mADB = ½mAB

A

D

B

130˚

(½)130 = 65˚

Ex. 1: Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle.

Q

RS

T

QTSm = 2mQRS =

2(90°) = 180°

m = 2mZYX =

Ex. 2: Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle.

ZWX

2(115°) = 230°

XY

Z

W

115˚

m = ½ m

Ex. 3: Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle.NMP

½ (100°) = 50°

NP

P

N

M

100°

Theorem 10.8

• If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

• C D

C

B

A

D

Ex. 4: Finding the Measure of an Angle

• It is given that mE = 75°. What is mF?

• E and F both intercept , so E F. So, mF = mE = 75°

GH

H

G

E

F

75°

Example

68/2 = 34

180-118 = 62

62/2 = 31

62 + 68 = 130

180-68 = 118 118 + 62 = 180

Same as arc QP=62

68 + 62 + 118 = 248

Assignment

• Practice Workbookp. 193(1-15)