10-2 Angles and Arcs10-2 Angles and Arcs10-2 Angles and Arcs10-2 Angles and Arcs

Central Angle• A central angle is an angle whose

vertex is at the center of a circle.

Sum of Central Angles• The sum of the measures of the

central angles of a circle with no interior points in common is 360.

Arc• An arc is an unbroken part of a

circle.

Minor Arc• Part of a circle that measures less

than 180°.central angle

minorarcmajor

arcP

B

A

C

Semicircle• An arc whose endpoints are the

endpoints of a diameter of the circle.

EH F

G

E

Major Arc• Part of a circle that measures

between 180° and 360°.central angle

minorarcmajor

arcP

B

A

C

Definition of Arc Measure

• The measure of a minor arc is the measure of its central angle. The measure of a major arc is 360 minus the measure of its central angle. The measure of a semicircle is 180.

Naming Arcs• Arcs are named by

their endpoints. For example, the minor arc associated with APB above is . Major arcs and semicircles are named by their endpoints and by a point on the arc.

AB

central angle

minorarcmajor

arcP

B

A

C

AB

Using Arcs of Circles• In a plane, an angle

whose vertex is the center of a circle is a central angle of the circle. If the measure of a central angle, APB is less than 180°, then A and B and the points of P

central angle

minorarcmajor

arcP

B

A

C

Using Arcs of Circles• The interior of APB

form a minor arc of the circle. The points A and B and the points of P in the exterior of ACB form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle.

central angle

minorarcmajor

arcP

B

A

C

Naming Arcs• For example, the

major arc associated with APB is .

The measure of a

minor arc is defined to be the measure of its central angle.

60°ACB

central angle

minorarcmajor

arcP

B

A

C

Naming Arcs• For instance, m

= mGHF = 60°. • m is read “the

measure of arc GF.” You can write the measure of an arc next to the arc. The measure of a semicircle is always 180°.

EH F

G

E

60°

60°

180°

GF

GF

Ex. 1: Finding Measures of Arcs• Find the

measure of each arc of R.

a. b. c.

MN

MPN

PMN PR

M

N80°

Adjacent Arcs• Adjacent arcs are arcs of a circle

that have exactly one point in common.

Note:• Two arcs of the same

circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas.

• Postulate 26—Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

B

A

C

Arc Addition Postulate• The measure of an arc formed by

two adjacent arcs is the sum of the measures of the two arcs.

Ex. 2: Finding Measures of Arcs

• Find the measure of each arc.

a. b. c.

GE

R

EF

G

H

GEFGF

40°

80°

110°

Arc Length• A portion of the circumference of a

circle.

Arc Length Formula• Arc length AB = mAB • 2лr 360°

R

EF

G

H

80

Find the arc length of HE and FE.

R

EF

G

H

75110

4 in

Concentric Circles• Concentric circles are circles that

have a common center.• Concentric circles lie in the same

plane and have the same center, but have different radii.

• All circles are similar circles.

Congruent Circles• Circles that have the same radius

are congruent circles.

Congruent Arcs• If two arcs of one circle have the

same measure, then they are congruent arcs.

• Congruent arcs also have the same arc length.

• Assignment page 710

• Class work 1-23 (turn in)• Homework 26-41