This document defines and discusses key concepts related to measuring angles and arcs in circles. It begins by introducing central angles, major arcs, minor arcs, semicircles and their measures. It then defines congruent arcs and adjacent arcs. The document provides examples of identifying different types of arcs and calculating their measures using properties such as the angles of a circle summing to 360 degrees and the arc measure being equal to the central angle measure for minor arcs. It also demonstrates calculating arc lengths using the formula. The goal is to understand how to identify and find measures of different angles and arcs in circles.

1. Section 10-2
Measuring Angles and Arcs
Monday, May 14, 2012

2. Essential Questions
• How do you identify central angles,
major arcs, minor arcs, and semicircles,
and find their measures?
• How do you find arc length?
Monday, May 14, 2012

3. Vocabulary
1. Central Angle:
2. Arc:
3. Minor Arc:
4. Major Arc:
Monday, May 14, 2012

4. Vocabulary
1. Central Angle: An angle inside a circle with the
vertex at the center and each side is a radius
2. Arc:
3. Minor Arc:
4. Major Arc:
Monday, May 14, 2012

5. Vocabulary
1. Central Angle: An angle inside a circle with the
vertex at the center and each side is a radius
2. Arc: A part of a exterior of the circle
3. Minor Arc:
4. Major Arc:
Monday, May 14, 2012

6. Vocabulary
1. Central Angle: An angle inside a circle with the
vertex at the center and each side is a radius
2. Arc: A part of a exterior of the circle
3. Minor Arc: An arc that is less than half of a
circle; Has same measure as the central angle
that contains it
4. Major Arc:
Monday, May 14, 2012

7. Vocabulary
1. Central Angle: An angle inside a circle with the
vertex at the center and each side is a radius
2. Arc: A part of a exterior of the circle
3. Minor Arc: An arc that is less than half of a
circle; Has same measure as the central angle
that contains it
4. Major Arc: An arc that is more than half of a
circle; Find the measure by subtracting the measure
of the minor arc with same length from 360°
Monday, May 14, 2012

8. Vocabulary
5. Semicircle:
6. Congruent Arcs:
7. Adjacent Arcs:
Monday, May 14, 2012

9. Vocabulary
5. Semicircle: An arc that is half of a circle; the
measure of a semicircle is 360°
6. Congruent Arcs:
7. Adjacent Arcs:
Monday, May 14, 2012

10. Vocabulary
5. Semicircle: An arc that is half of a circle; the
measure of a semicircle is 360°
6. Congruent Arcs: Arcs that have the same measure
7. Adjacent Arcs:
Monday, May 14, 2012

11. Vocabulary
5. Semicircle: An arc that is half of a circle; the
measure of a semicircle is 360°
6. Congruent Arcs: Arcs that have the same measure
7. Adjacent Arcs: Two arcs in a circle that have
exactly one point in common
Monday, May 14, 2012

12. Theorems and Postulates
Theorem 10.1 - Congruent Arcs:
Postulate 10.1 - Arc Addition Postulate:
Arc Length:
Monday, May 14, 2012

13. Theorems and Postulates
Theorem 10.1 - Congruent Arcs: In the same or
congruent circles, two minor arcs are congruent
IFF their central angles are congruent
Postulate 10.1 - Arc Addition Postulate:
Arc Length:
Monday, May 14, 2012

14. Theorems and Postulates
Theorem 10.1 - Congruent Arcs: In the same or
congruent circles, two minor arcs are congruent
IFF their central angles are congruent
Postulate 10.1 - Arc Addition Postulate: The
measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs
Arc Length:
Monday, May 14, 2012

15. Theorems and Postulates
Theorem 10.1 - Congruent Arcs: In the same or
congruent circles, two minor arcs are congruent
IFF their central angles are congruent
Postulate 10.1 - Arc Addition Postulate: The
measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs
D
Arc Length: l = i2π r
360
Monday, May 14, 2012

16. Example 1
Find the value of x when m∠QTV = (20x)°,
m∠QTR = 20°, m∠RTS = (8x − 4)°,
m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°.
Monday, May 14, 2012

17. Example 1
Find the value of x when m∠QTV = (20x)°,
m∠QTR = 20°, m∠RTS = (8x − 4)°,
m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°.
20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360
Monday, May 14, 2012

18. Example 1
Find the value of x when m∠QTV = (20x)°,
m∠QTR = 20°, m∠RTS = (8x − 4)°,
m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°.
20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360
46x + 38 = 360
Monday, May 14, 2012

19. Example 1
Find the value of x when m∠QTV = (20x)°,
m∠QTR = 20°, m∠RTS = (8x − 4)°,
m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°.
20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360
46x + 38 = 360
46x = 322
Monday, May 14, 2012

20. Example 1
Find the value of x when m∠QTV = (20x)°,
m∠QTR = 20°, m∠RTS = (8x − 4)°,
m∠STU = (13x − 3)°, and m∠VTU = (5x + 5)°.
20x + 40 + 8x − 4 + 13x − 3 + 5x + 5 = 360
46x + 38 = 360
46x = 322
x=7
Monday, May 14, 2012

21. Example 2
WC is the radius of ⊙C. Identify each as a major arc,
minor arc, or semicircle. Then find each measure.

a. XZY

b. WZX

c. XW
Monday, May 14, 2012

22. Example 2
WC is the radius of ⊙C. Identify each as a major arc,
minor arc, or semicircle. Then find each measure.

a. XZY Semicircle, 180°

b. WZX

c. XW
Monday, May 14, 2012

23. Example 2
WC is the radius of ⊙C. Identify each as a major arc,
minor arc, or semicircle. Then find each measure.

a. XZY Semicircle, 180°

b. WZX Major arc, 270°

c. XW
Monday, May 14, 2012

24. Example 2
WC is the radius of ⊙C. Identify each as a major arc,
minor arc, or semicircle. Then find each measure.

a. XZY Semicircle, 180°

b. WZX Major arc, 270°

c. XW Minor arc, 90°
Monday, May 14, 2012

25. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
a. Find the measure of the arc of Comfort
the section that represents the 21%
Youth Hybrid
comfort bicycles. 26% 9%
Other
7%
Mountain
37%
Monday, May 14, 2012

26. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
a. Find the measure of the arc of Comfort
the section that represents the 21%
Youth Hybrid
comfort bicycles. 26% 9%
Other
7%
360(.21) Mountain
37%
Monday, May 14, 2012

27. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
a. Find the measure of the arc of Comfort
the section that represents the 21%
Youth Hybrid
comfort bicycles. 26% 9%
Other
7%
360(.21 = 75.6°
) Mountain
37%
Monday, May 14, 2012

28. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
b. Find the measure of the arc Comfort
representing the combination of 21%
Youth Hybrid
the mountain, youth, and comfort 26% 9%
bicycles. Other
7%
Mountain
37%
Monday, May 14, 2012

29. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
b. Find the measure of the arc Comfort
representing the combination of 21%
Youth Hybrid
the mountain, youth, and comfort 26% 9%
bicycles. Other
7%
360(.37 + .26 + .21) Mountain
37%
Monday, May 14, 2012

30. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
b. Find the measure of the arc Comfort
representing the combination of 21%
Youth Hybrid
the mountain, youth, and comfort 26% 9%
bicycles. Other
7%
360(.37 + .26 + .21 = 360(.84)
) Mountain
37%
Monday, May 14, 2012

31. Example 3
Refer to the table showing the percent of bicycles
bought by type at a bike shop.
Type Mountain Youth Comfort Hybrid Other
Percent 37% 26% 21% 9% 7%
b. Find the measure of the arc Comfort
representing the combination of 21%
Youth Hybrid
the mountain, youth, and comfort 26% 9%
bicycles. Other
7%
360(.37 + .26 + .21 = 360(.84)
) Mountain
37%
= 302.4°
Monday, May 14, 2012

32. Example 4
Find the measure of each arc.

a. mKHL

b. mHJ

c. mLH

d. mKJ
Monday, May 14, 2012

33. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32

b. mHJ

c. mLH

d. mKJ
Monday, May 14, 2012

34. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32 = 328°

b. mHJ

c. mLH

d. mKJ
Monday, May 14, 2012

35. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32 = 328°

b. mHJ = 180 − 32

c. mLH

d. mKJ
Monday, May 14, 2012

36. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32 = 328°

b. mHJ = 180 − 32 = 148°

c. mLH

d. mKJ
Monday, May 14, 2012

37. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32 = 328°

b. mHJ = 180 − 32 = 148°

c. mLH = 32°

d. mKJ
Monday, May 14, 2012

38. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32 = 328°

b. mHJ = 180 − 32 = 148°

c. mLH = 32°

d. mKJ = 90 + 32
Monday, May 14, 2012

39. Example 4
Find the measure of each arc.

a. mKHL = 360 − 32 = 328°

b. mHJ = 180 − 32 = 148°

c. mLH = 32°

d. mKJ = 90 + 32 = 122°
Monday, May 14, 2012

40. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
a.
Monday, May 14, 2012

41. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
a. D
l= i2π r
360
Monday, May 14, 2012

42. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
a. D
l= i2π r
360
40
= i2π (4.5)
360
Monday, May 14, 2012

43. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
a. D
l= i2π r
360
40
= i2π (4.5)
360
≈ 3.14 cm
Monday, May 14, 2012

44. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
b.
Monday, May 14, 2012

45. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
b. D
l= i2π r
360
Monday, May 14, 2012

46. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
b. D
l= i2π r
360
152
= i2π (6)
360
Monday, May 14, 2012

47. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
b. D
l= i2π r
360
152
= i2π (6)
360
≈ 15.92 cm
Monday, May 14, 2012

48. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
c.
Monday, May 14, 2012

49. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
c. D
l= i2π r
360
Monday, May 14, 2012

50. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
c. D
l= i2π r
360
140
= i2π (6)
360
Monday, May 14, 2012

51. Example 5

Find the length of DA ,
rounding to the nearest hundredth.
c. D
l= i2π r
360
140
= i2π (6)
360
≈ 14.66 cm
Monday, May 14, 2012

52. Check Your Understanding
p. 696 #1-11
Monday, May 14, 2012

53. Problem Set
Monday, May 14, 2012

54. Problem Set
p. 696 #13-41 odd, 55, 73
"Our lives improve only when we take chances - and
the first and most difficult risk we can take is to be
honest with ourselves."
- Walter Anderson
Monday, May 14, 2012