9.4 Inscribed Angles

Geometry

Objectives/Assignment

• Use inscribed angles to solve problems.

• Use properties of inscribed polygons.

Review

Definitions• An inscribed

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

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

intercepted arc

inscribed angle

Theorem 9.4: Measure of an Inscribed Angle

m ADB = ½mAB

C

B

A

D

Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle.

Q

RS

T

QTSm = 2mQRS =

2(90°) = 180°

Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle if ZYX = 115 °.

ZWXm = 2mZYX

2(115°) = 230°X

Y

Z

W

Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle.NMPm = ½ m

½ (100°) = 50°

NP

P

N

M

100°

Theorem 9.5

• C D

C

B

A

D

Comparing Measures of Inscribed Angles• Find mACB,

mADB, and mAEB if AB = 60 °.

The measure of each angle is half the measure of

m = 60°, so the measure of each angle is 30°

AB

E

D C

B

A

AB

Finding the Measure of an Angle

• Given m E = 75°. What is m F?

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

GH

H

G

E

F

75°

• Find x.

• AB is a diameter. So, C is a right angle and mC = 90°

• 2x° = 90°• x = 45

A

Q

C

B

2x°

• ½ * 96 = (2x + 1)• 48 = 2x + 1• 47 = 2x• X = 23.5

m PQR = ½ m PR

• x = 2x – 3• - x = -3• X = 3

• m P = 90 • ½ x + (1/3 x +5) = 90• 5/6 x + 5 = 90• 5/6 x = 85• X = 102

Practice