1. Angles and Arcs angles and their measures. Find arc length.
Recognize major arcs, minor arcs, semicircles, and central
angles and their measures.
Find arc length.
An artist’s rendering of Larry Niven’s Ringworld.

2. Two artist’s conceptions of views from the surface of Ringworld.

3. ANGLES AND ARCS
A central angle has the center of a circle as its vertex, and its sides contain two radii of the circle.
The sum of the measures of the angles around the center of a circle is 360°.

4. Key Concept Sum of Central Angles
The sum of the measures of the central angles of a circle with no interior points in common is 360°. 1 3 2

5. Example 1 Measure of Central Angles
a) Find mAOB B
3x° C
25x°
2x° A D O E
b) Find mAOE

6. Key Concept Arcs of a Circle
Minor Arc
Major Arc
Semicircle A D K E J
60°
110° C B G N L F M
Usually named using the letters of the two endpoints.
Named by the letters of the two endpoints and another point on the arc.
Named by the letters of the two endpoints and another point on the arc. AC
DFE
JML and JKL

7. Key Concept Arcs of a Circle
Minor Arc
Major Arc
Semicircle
JKL A D
DFE AC K E J
60°
110° C B G N L F M
JML
Usually named using the letters of the two endpoints.
Named by the letters of the two endpoints and another point on the arc.
Named by the letters of the two endpoints and another point on the arc. AC
DFE
JML and JKL

8. Theorem
In the same or in congruent circles, two arcs are congruent if and only if their corresponding central angles are congruent.

9. Postulate Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. P S Q R
In circle S, mPQ + mQR = mPQR

10. Example 2 Measures of Arcs
a) Find mBE D
50° A B F
b) Find mCBE E
c) Find mACE

11. Example 3 Circle Graphs BICYCLES
This graph shows the percent of each type of bicycle sold in the United States in 2001.
Mountain
37%
133.2°
Youth
26%
93.6°
Hybrid 8%
28.8°
Other
Comfort
21%
75.6°
Identify any arcs that are congruent.

12. ARC LENGTH Another way to measure an arc is by its length.
An arc is part of the circle, so the length of an arc is part of the circumference.

13. Example 4 Arc Length
In circle P, PR = 15 and mQPR = 120°. Find the length of arc QR. Q
Solution:
r = 15, so C = 215 or 30, and
arc QR = mQPR or 120°.
Write a proportion to compare each part to its whole.
120° P 15 R
l  units

14. Key Concept Arc Length
The proportion in the last example can be adapted to find an arc length in any circle.
degree measure of an arc 
degree measure of whole circle 
arc length
circumference
This can also be expressed as