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Lesson 2Circles

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Arcs

• An arc is an unbroken part of a circle.

• For example, in the figure, the part of the circle shaded red is an arc.

• A semicircle is an arc equal to half a circle.

• A minor arc is smaller than a semicircle.

• A major arc is larger than a semicircle.

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Naming Arcs

• A minor arc, like the one in red in the figure, can be named by drawing an arc symbol over its endpoints:

• Sometimes, to avoid confusion, a third point between the endpoints is used to name the arc:

A

B

P or .AB BA

or .APB BPA

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• Semicircles and major arcs must be named with three (or sometimes more) points.

• The arc highlighted in red in the figure would be called

It appears to be a major arc.

A

B

C

or .ABC CBA

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The Measure of an Arc• Each arc has a degree measure between 0 degrees and 360 degrees.• A full circle is 360 degrees, a semicircle is 180 degrees, a minor arc measures less

than 180 degrees, and a major arc measures more than 180 degrees.• If an arc is a certain fraction of a circle, then its measure is the same fraction of 360

degrees. Some sample arc measures are given below.

C

A

E

B

DF

G

H

45mAB 90mABC 135mABD 180mACE 225mADF 270mADG

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Central Angles

• Given a circle, a central angle is an angle whose vertex is at the center of the circle.

• In the figure, the center of the circle is and is a central angle that intercepts arc

• The measure of a central angle is equal to the measure of the arc it intercepts.

P

AQ

B

P APB

.AQB

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Inscribed Angles

• In a circle, an inscribed angle is an angle whose vertex is on the circle and whose sides are chords.

• In the figure, is an inscribed angle and it intercepts arc

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

P

A

B

QAPB

.AQB

Inscribed Angles

•The inscribed angles subtended by the same arc are congruent.

Central and Inscribed Angles

•The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc.

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Example

• In the figure, P is the center of the circle and

• Find and

A

PB

C

D

30 .m A

, , , m B m CPD m BCD .m PCD

30

60

6060

Since inscribed intercepts ,

2 60 .

A CD

mCD m A

Then, since intercepts the

same arc, 0.5 60 30 .

B

m B

Then, since is a central angle

intercepting CD, 60 .

CPD

m CPD mCD

Since is inscribed in a semicircle,

90 .

BCD

m BCD

Since is isosceles with vertex ,

0.5(180 60 ) 60 .

CPD P

m PCD