Geo

met

ry 10.2 Finding Arc Measures2/24/2010

Geo

met

ryObjectives/Assignment

• Use properties of arcs of circles, as applied.

Geo

met

ryUsing 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 are the points of P

central angle

minorarcmajor

arcP

B

A

C

Geo

met

ryUsing 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 APB 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

• Minor Arc: < 180˚• Major Arc: > 180˚

Geo

met

ryNaming Arcs

• Arcs are named by their endpoints.

• Major arcs and semicircles are named by their endpoints and by a point on the arc. (3 Letters).

AB

central angle

minorarcmajor

arcP

B

A

C

ACB

• The major arc is named

• The minor arc is named

Geo

met

ryNaming Arcs• Name the semicircle.

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

EH F

G

E

60°

60°

Geo

met

ryNaming 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

Geo

met

ryNaming Arcs• The measure of the

whole circle is 360°.

• The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc.

• Find m .

EH F

G

E

60°

60°

180°

GEF

Geo

met

ryEx. 1: Finding Measures of Arcs

• Find the measure of each arc of R.

a. b. c.

MNMPN

PMN PR

M

N 80°

Geo

met

ryEx. 1: Finding Measures of Arcs

• Find the measure of each arc of R.

a. b. c. Solution:

is a minor arc, so m = mMRN = 80°

MNMPN

PMN PR

M

N 80°

MN

MN

Geo

met

ryEx. 1: Finding Measures of Arcs

• Find the measure of each arc of R.

a. b. c. Solution:

is a major arc, so m = 360° – 80° = 280°

MNMPN

PMN PR

M

N 80°

MPN

MPN

Geo

met

ryEx. 1: Finding Measures of Arcs

• Find the measure of each arc of R.

a. b. c. Solution:

is a semicircle, so m = 180°

MNMPN

PMN PR

M

N 80°

PMN

PMN

Geo

met

ryNote:• 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.

ABC

B

A

C

m = m + m AB

BC

Geo

met

ryEx. 2: Finding Measures of Arcs

• Find the measure of each arc.

a. b. c. m = m + m =40° + 80° = 120°

GE

R

EF

G

H

GEFGFGE

GH

HE

40°

80°

110°

Geo

met

ryEx. 2: Finding Measures of Arcs

• Find the measure of each arc.

a. b. c. m = m + m =120° + 110° = 230°

GE

R

EF

G

H

GEFGF

EF

40°

80°

110°GEF

GE

Geo

met

ryEx. 2: Finding Measures of Arcs

• Find the measure of each arc.

a. b. c. m = 360° - m =360° - 230° = 130°

GE

R

EF

G

H

GEFGF

40°

80°

110°GF

GEF