12.3 Inscribed Angles12.3 Inscribed Angles

Using Inscribed Angles• 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 12.9: 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

C

B

A

D

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°

Ex. 1: Finding Measures of Arcs and Inscribed Angles

• Find the measure of the blue arc or angle.

ZWXm = 2mZYX =

2(115°) = 230°X

Y

Z

W

Ex. 1: 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°

Ex. 2: Comparing Measures of Inscribed Angles

• Find mACB, mADB, and mAEB.

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

Corollaries to Th. 12-9

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

C DC

B

A

D

Ex. 3: 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°

• A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

• D, E, F, and G lie on some circle, C, if and only if mD + mF = 180° and mE + mG = 180°

C

G

F

E

D

• Find the value of each variable.

• DEFG is inscribed in a circle, so opposite angles are supplementary.

• mD + mF = 180°• z + 80 = 180• z = 100

F

ED

G

120°

80°

y°

z°

Ex. 5:

• Find the value of each variable.

• DEFG is inscribed in a circle, so opposite angles are supplementary.

• mE + mG = 180°• y + 120 = 180• y = 60

F

ED

G

120°

80°

y°

z°

• If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

B is a right angle if and only if AC is a diameter of the circle.

A

C

B

• Find the value of each variable.

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

• 2x° = 90°• x = 45

A

Q

C

B

2x°

Find each measure.

mEFH

= 65°