10-4 Inscribed Angles

You found measures of interior angles of polygons.

• Find measures of inscribed angles.

• Find measures of angles of inscribed polygons.

Find the measure of each arc

• mDE

• mAED

• mEBDA

B

C

D

E

54°

54°

180°

306°

What kind of angle is angle ECD?

Central angle

DefinitionHis name was inscribed on the award.

The square is inscribed in the circle.

A

B

CAn inscribed angle is an angle with its vertex on a circle and sides that contain chords of the circle.

Intercepted ArcAn intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle

Q

S

C

R

Intercepted arc

P P P

Center P is on a side of the inscribed angle

Center P is inside the inscribed angle

Center P is in the exterior of the inscribed angle

Sizing Up Inscribed Angles

• Measure the central angle

• What is the measure of the arc AB?

• Measure the inscribed angle

• Compare the measure of the central angle and the inscribed angle.

A

BC

D

ACB

ADB

p. 723

The measure of an inscribed angle is half the measure of its intercepted arc.

A

B

C

p. 724

Find the measure of each angle or arc indicated by a variable.

24°

x°

160°

y°

C

z°

48°

80°

90°

A. Find mX.

Answer: mX = 43

B.

= 2(52) or 104

A. 47

B. 54

C. 94

D. 188

A. Find mC.

If two inscribed angles intercept the same arc, then they are congruent.

A

B

C

D

p. 724

ALGEBRA Find mR.

R S R and S both intercept . mR mS Definition of congruent angles

12x – 13 = 9x + 2 Substitutionx = 5 Simplify.

Answer: So, mR = 12(5) – 13 or 47.

A. 4

B. 25

C. 41

D. 49

ALGEBRA Find mI.

An inscribed angle that intercepts a semicircle is a right angle.

p. 725

ALGEBRA Find mB.

ΔABC is a right triangle because C inscribes a semicircle.

mA + mB + mC = 180 Angle Sum Theorem(x + 4) + (8x – 4) + 90 = 180 Substitution

9x + 90 = 180 Simplify.9x = 90 Subtract 90 from each

side.x = 10 Divide each side by 9.

Answer: So, mB = 8(10) – 4 or 76.

10-4 Assignment

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