Inscribed AnglesInscribed AnglesInscribed AnglesInscribed Angles

10-410-4

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

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 = ½m

AB

C

B

A

D

Ex. 1: Finding Measures of Arcs and Inscribed Angles• Find the measure

of the blue arc or angle.

M<NMP = ½ NP P

N

M

100°

ExampleExample

A

B

C

D

70o

7xo

Find x.

Theorem 9-5• If two inscribed angles of a circle

or congruent circles intercept congruent arcs or the same arc, then the angles are congruent.

Comparing Measures of Inscribed Angles

E

D C

B

A

Ex. 3: Finding the Measure of an Angle• It is given that

mE = 75°. What is mF?

H

G

E

F

75°

Theorem 9-6• If an inscribed angle of a circle

intercepts a semicircle, then the angle is a right angle.

ExampleExample

A

B

CD

E

m AED= ?(

Ex. 1: Finding Measures of Arcs and Inscribed Angles• Find the measure

of the blue arc or angle.

Q

RS

T

• Find the value of each variable.

A

Q

C

B

2x°

Ex: find x.Ex: find x.P

C

Q

R3xo

Theorem 9-7• If a quadrilateral is inscribed in a

circle, then its opposite angles are supplementary.

• 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.

F

ED

G

120°

80°

y°

z°

That’s all folks!

• Class work • Page 727 problems 1-6, 8-20• Homework• Page 728 problems 28-30