1. Arcs and Central Angles
Section 9-3
Arcs and Central Angles

2. 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

3. Arcs An arc is an unbroken section of a circle m AB = 88˚
The measure of an arc is equal to the measure of its central angle A B
88˚
m AB = 88˚
A circle has a total of 360˚

4. 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 semicircle D B G E A F H AB
FGH
CDE C

5. It’s like…
A circle that has been cut in half.

6. Example 1 C If CA is a diameter of circle O: Name: Two minor arcs
Two major arcs
Two semicircles
An acute central angle
Two arcs with the same measure R O A S

7. Adjacent Arcs
Adjacent arcs are arcs that have exactly one endpoint in common and share no other points B C A

8. 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

9. Example 2
Give the measure of each angle or arc if YT is a diameter of circle O WX
<WOT
XYT Y X
30˚ Z O W
50˚ T

10. 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

11. 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
75˚ O
8 cm F
8 cm D
75˚ I

12. 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.
Conclusion:
m<1 = m<2 P B 2
75˚ 1 T F
8 cm
8 cm A
75˚ I

13. Example 3 Find the measure of central <1 72˚ 95˚ 150˚ 50˚ 1.) 2.)
40˚ 1 1
72˚
225˚
3.)
150˚
4.)
50˚ 1 1
130˚
30˚

14. Example 4 D Find the measure of each arc. 1.) AB 2.) BC 3.) CD 4.) DE

15. Or is it?
The
End