10-2 angles and arcs
DESCRIPTION
10-2 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. - PowerPoint PPT PresentationTRANSCRIPT
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