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Angles in a Circle

Chapter 7 The Circle

7.3B

7.3.1

MATHPOWERTM 11, WESTERN EDITION

Angles in a Circle Theorems

1. The measure of the central angle is equal

to twice the measure of the inscribed

angle subtended by the same arc. 2.

Inscribed angles subtended by the same arc,

or by equal arcs, are congruent. 3. The

angle inscribed in a semicircle is a right angle.

C

O

Both angles are subtended by the arc AB.

B

A

7.3.2

Measure of the Arcs

The arc AB will have the same measure as that of

the central angle AOB.

O

550

400

700

A

B

700

400

550

C

The arc AB will be twice the measure of

the inscribed angle ACB.

300

400

B

A

7.3.3

600

800

Angles in a Circle - Theorem 1

The measure of the central angle is equal to

twice the measure of the inscribed angle

subtended by the same arc.

B

Inscribed Angle

Central Angle

Radii

AO BO

O

ITT

Exterior Angle Theorem

A

Substitution

C

Simplify

Prove that

7.3.4

Angles in a Circle - Theorem 2

Inscribed angles subtended by the same arc, or

by equal arcs, are congruent.

C

B

O

D

Angles in a Circle

A

Prove that

Angles in a Circle

Transitive

7.B.5

Angles in a Circle - Theorem 3

The angle inscribed in a semicircle is a right

angle.

C

B

Angles in a circle

O

Straight Angle

A

Substitution

Prove

7.3.6

Using the Angles in a Circle Theorems

Find the measure of the indicated angles.

1580

Note All of the angles are subtended by the

arc EF. The measure of arc EF 860.

D

430

B

A

430

O

430

580

580

860

Given more information.

790

E

F

790

860

7.3.7

Using the Angles in a Circle Theorems

A

B

C

900

7.3.8

Using the Angles in a Circle Theorems

900

B

200

620

640

C

590

O

310

400

A

390

580

D

E

620

7.3.9

Assignment

Suggested Questions

Pages 412 and 413 1-24, 29

7.3.10

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