1. 10.2 Measuring
Angles and Arcs
Reitz High School

2. Targets:
Recognize major arcs, minor arcs, semicircles, and central angles and their measures
Find arc length

3. Central Angle
An angle with its vertex located at the center of a circle
THE MEASURE OF A CENTRAL ANGLE IS THE SAME AS THE MEASURE OF ITS INTERCEPTED ARC.

4. Central Angle AOB.

5. Angles and Arcs The sum of the measures of the central angles is 360°.
m<1 + m<2 +m<3 +m<4 = 360

6. Minor Arc
A minor arc is less then 180° and is labeled using the two endpoints.
A major arc is greater than 180° but less than 360° and is labeled using the two endpoints and another point on the arc.

7. Minor Arc : Label with 2 endpoints: UV or VU

8. Major Arc: Label with 3 points: ACB

9. Semicircle
A semicircle measures 180° and is labeled using the two endpoints and another point on the arc.

10. Semicircle: Label with 3 letters: AKB, ACB, AHB

11. Angles and Arcs
Theorem 10.1: In the same circle or  circles, two arcs are  iff their corresponding central angles are .

12. Postulate 10.1: Arc Addition:
Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the arcs.

13. Arc Addition Sketch:

14. Example 1a:
ALGEBRA: Refer to Assume RV is a diameter.
Find

15. Example 1a: The sum of the measures of Substitution Simplify.
Add 2 to each side.
Divide each side by 26.
Use the value of x to find
Given
Substitution
Answer: 52

16. Example 1b:
ALGEBRA: Refer to Assume RV is a diameter.
Find

17. Example 1b: form a linear pair. Linear pairs are supplementary.
Substitution
Simplify.
Subtract 140 from each side.
Answer: 40

18. Your Turn: Refer to . Assume AD and BE are diameters. a. Find m
b. Find m
Answer: 65
Answer: 40

19. Example 2a:
In bisects and
Find

20. Example 2a: is a minor arc, so is a semicircle. is a right angle.
Arc Addition Postulate
Substitution
Subtract 90 from each side.
Answer: 90

21. Example 2b:
In bisects and
Find

22. Example 2b: since bisects . is a semicircle. Arc Addition Postulate
Subtract 46 from each side.
Answer: 67

23. Example 2c:
In bisects and
Find

24. Example 2c: Vertical angles are congruent. Substitution. Substitution.
Subtract 46 from each side.
Substitution.
Subtract 44 from each side.
Answer: 316

25. Your Turn: In and are diameters, and bisects Find each measure. a. b.
Answer: 54
Answer: 72
Answer: 234

26. Arc Length
Another way to measure an arc is by its length. An arc is part of a circle, so its length is part of the circumference. Arc Length = 𝑚 ⫪r
m= measure of central angle
r= radius of circle

27. Example 3:
In and Find the length of

28. Example 3: degree measure of arc degree measure of whole circle
arc length
circumference
Answer: The length of is units or about 3.14 units.

29. Your Turn: In and . Find the length of .
Answer: units or about units