Chapter 7 Chapter 7 Lesson 6Lesson 6

Objective:Objective: To find the To find the measures of central angles measures of central angles

and arcs.and arcs.

Centr

al

Centr

al

Angles

Angles

and A

rcs

and A

rcs

• In a plane, a In a plane, a circlecircle is the set of all points. is the set of all points.

• The set of all points equidistant from a given point is the The set of all points equidistant from a given point is the centercenter..

• A A radiusradius is a segment that has one endpoint at the is a segment that has one endpoint at the center and the other endpoint on the circle.center and the other endpoint on the circle.

• A A diameterdiameter is a segment that contains the center of a is a segment that contains the center of a circle and has both endpoints on the circle.circle and has both endpoints on the circle.

Congruent CirclesCongruent Circles have congruent radii. have congruent radii.

5 m

5 m

5 m

5 m

Central Angle is an angle whose vertex is the center of the circle. AA

BBCCDDABD ABC

Example 1Finding Central Angles

**Remember a circle measures **Remember a circle measures 360360°.**°.**Sleep:Sleep: 31% of 360

.31•360=111.6

FoodFood:: 9% of 360 .09•360=32.4Work:Work: 20% of 360 .20•360=72

Must DoMust Do:: 7% of 360 .07•360=25.2Entertainment:Entertainment: 18% of 360 .18•360=64.8

Other:Other: 15% of 360 .15•360=54

• An arcarc is a part of a circle. Types of arcsTypes of arcs

• Semicircle is half of a circle.

                                                                                                                                                                                          

                          

•AA

DAEDAE

• A minor arc is smaller than a semicircle.

• A major arc is greater than a semicircle.

ABAB

Minor arcMinor arc

•DD ADBADB

Major arcMajor arc

Example 2:Identifying Arcs

Identify the following in O.  

1.the minor arcs

2.the semicircles

3. the major arcs that contain point A

•OOAA CC

DD EE

Example 3:Identifying Arcs

Identify the minor arcs, major arcs and semicircles in O with point A as an endpoint.  

O••

•

•

•

A

B

D

E

• minor arcs

AD, AE

• major arcs

ADE, AED

• semicircles

ADB, AEB

Adjacent arcs are arcs of the same circle that have exactly one point in common.

Postulate 7-1Postulate 7-1: Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

•• •

AA

BB C

mABC = mAB + mBC

Example 4:Finding the Measures of Arcs

Find the measure of each arc.Find the measure of each arc.

•

58°

32°

A

B

CD

O

• BC

mBOCmBC 32

• BD

180mABC

5832mBD 90• ABC

ABC is a semicircle.

mCDmBCmBD

• AB 32180mAB 148

Example 5:Finding the Measures of Arcs

56°

40°

M

C W

XD

Y

Find mXY and mDXM in C.

mXY = mXD + mDY

mXY = 40 + 56 = 96

mDXM = mDX + 180

mDXM = 40 + 180

mDXM = 220

AssignmentAssignment

pg.389-392 #1-26; 40-54