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

2. Essential Questions How do you identify central angles, major arcs, minor arcs, and semicircles, and nd their measures? How do you nd 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 360Monday, 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 commonMonday, 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 = i2 r 360Monday, May 14, 2012

16. Example 1 Find the value of x when mQTV = (20x), mQTR = 20, mRTS = (8x 4), mSTU = (13x 3), and mVTU = (5x + 5).Monday, May 14, 2012

17. Example 1 Find the value of x when mQTV = (20x), mQTR = 20, mRTS = (8x 4), mSTU = (13x 3), and mVTU = (5x + 5). 20x + 40 + 8x 4 + 13x 3 + 5x + 5 = 360Monday, May 14, 2012

18. Example 1 Find the value of x when mQTV = (20x), mQTR = 20, mRTS = (8x 4), mSTU = (13x 3), and mVTU = (5x + 5). 20x + 40 + 8x 4 + 13x 3 + 5x + 5 = 360 46x + 38 = 360Monday, May 14, 2012

19. Example 1 Find the value of x when mQTV = (20x), mQTR = 20, mRTS = (8x 4), mSTU = (13x 3), and mVTU = (5x + 5). 20x + 40 + 8x 4 + 13x 3 + 5x + 5 = 360 46x + 38 = 360 46x = 322Monday, May 14, 2012

20. Example 1 Find the value of x when mQTV = (20x), mQTR = 20, mRTS = (8x 4), mSTU = (13x 3), and mVTU = (5x + 5). 20x + 40 + 8x 4 + 13x 3 + 5x + 5 = 360 46x + 38 = 360 46x = 322 x=7Monday, May 14, 2012

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

22. Example 2 WC is the radius of C. Identify each as a major arc, minor arc, or semicircle. Then nd each measure. a. XZY Semicircle, 180 b. WZX c. XWMonday, May 14, 2012

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

24. Example 2 WC is the radius of C. Identify each as a major arc, minor arc, or semicircle. Then nd each measure. a. XZY Semicircle, 180 b. WZX Major arc, 270 c. XW Minor arc, 90Monday, 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% = 302.4Monday, May 14, 2012

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

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

34. Example 4 Find the measure of each arc. a. mKHL = 360 32 = 328 b. mHJ c. mLH d. mKJMonday, May 14, 2012

35. Example 4 Find the measure of each arc. a. mKHL = 360 32 = 328 b. mHJ = 180 32 c. mLH d. mKJMonday, 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. mKJMonday, 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. mKJMonday, 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 + 32Monday, 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 = 122Monday, 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= i2 r 360Monday, May 14, 2012

42. Example 5 Find the length of DA , rounding to the nearest hundredth. a. D l= i2 r 360 40 = i2 (4.5) 360Monday, May 14, 2012

43. Example 5 Find the length of DA , rounding to the nearest hundredth. a. D l= i2 r 360 40 = i2 (4.5) 360 3.14 cmMonday, 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= i2 r 360Monday, May 14, 2012

46. Example 5 Find the length of DA , rounding to the nearest hundredth. b. D l= i2 r 360 152 = i2 (6) 360Monday, May 14, 2012

47. Example 5 Find the length of DA , rounding to the nearest hundredth. b. D l= i2 r 360 152 = i2 (6) 360 15.92 cmMonday, 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= i2 r 360Monday, May 14, 2012

50. Example 5 Find the length of DA , rounding to the nearest hundredth. c. D l= i2 r 360 140 = i2 (6) 360Monday, May 14, 2012

51. Example 5 Find the length of DA , rounding to the nearest hundredth. c. D l= i2 r 360 140 = i2 (6) 360 14.66 cmMonday, May 14, 2012

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

53. Problem SetMonday, May 14, 2012

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