Transcript Section 9-3

Section 9-3
Arcs and Central Angles
Central Angle

A central angle of a circle has its vertex in the
center of the circle
<1 is a central angle in
.
P
1
P
Arcs

An arc is an unbroken section of a circle
88˚
The measure of an arc is equal to
the measure of its central angle
m AB = 88˚
A circle has a total of 360˚
A
B
88˚
Minor and Major Arcs
A minor arc is less than 180˚
A major arc is more than 180˚
An arc that is exactly 180˚ is a
B semicircle
G
D
E
A
F
AB
H
FGH
C
CDE
It’s like…
A circle that has been cut in half.
Example 1
C
If CA is a diameter of circle O:
Name:
1. Two minor arcs
2. Two major arcs
O
R
3. Two semicircles
4. An acute central angle
5. Two arcs with the same measure
A
S
Adjacent Arcs

Adjacent arcs are arcs that have exactly one
endpoint in common and share no other points
B
C
A
Arc Addition Postulate (postulate 16)

The measure of the arc formed by two
adjacent arcs is the sum of the measures of
these arcs.
B
m AB + m BC = m ABC
64˚
140˚
Example:
C
m ABC = 204˚
A
Example 2
Give the measure of each angle or arc if YT is a diameter of circle O
1. WX
Y
X
2. <WOT
30˚
3. XYT
O
W
50˚
T
Z
Congruent Arcs

Two arcs are congruent if they have the same
measure, and if they lie on the same circle or
two congruent circles.
C
67˚
B
O
D
67˚
A
Congruent Arcs

Two arcs are congruent if they have the same
measure, and if they lie on the same circle or
two congruent circles.
P
B
F
8 cm
75˚
I
75˚
O
8 cm
D
Theorem 9-3

In the same circle or in congruent circles, two
minor arcs are congruent if and only if their
central angles are congruent.
P
Conclusion:
m<1 = m<2
B
1
F
T
8 cm
75˚
2
8 cm
A
75˚
I
Example 3
Find the measure of central <1
72˚
1.)
2.)
95˚
40˚
1
1
72˚
225˚
3.)
150˚
4.)
50˚
1
1
30˚
130˚
Example 4
D
C
Find the measure of each arc.
1.) AB
2.) BC
3.) CD
E
4x 2x
3x
4.) DE
5.) EA
A
B
Or is it?
The End