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# Geometry Section 10-6 1112

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Secants, Tangents, and Angle Measures

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### Geometry Section 10-6 1112

1. 1. Section 10-6 Secants, Tangents, and Angle MeasuresMonday, May 21, 2012
2. 2. Essential Questions How do you ﬁnd measures of angles formed by lines intersecting on or inside a circle? How do you ﬁnd measure of angles formed by lines intersecting outside the circle?Monday, May 21, 2012
3. 3. Vocabulary & Theorems 1. Secant: Theorem 10.12 - Two Secants:Monday, May 21, 2012
4. 4. Vocabulary & Theorems 1. Secant: A line that intersects a circle in exactly two points Theorem 10.12 - Two Secants:Monday, May 21, 2012
5. 5. Vocabulary & Theorems 1. Secant: A line that intersects a circle in exactly two points Theorem 10.12 - Two Secants: If two secants or chords intersect in the interior of a circle, then the measure of an angle formed is half of the sum of the measure of the arcs intercepted by the angle and its vertical angleMonday, May 21, 2012
6. 6. Vocabulary & Theorems Theorem 10.13 - Secant and Tangent:Monday, May 21, 2012
7. 7. Vocabulary & Theorems Theorem 10.13 - Secant and Tangent: If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is half of the measure of its intercepted arcMonday, May 21, 2012
8. 8. Vocabulary & Theorems Theorem 10.14 - Exterior Intersection:Monday, May 21, 2012
9. 9. Vocabulary & Theorems Theorem 10.14 - Exterior Intersection: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcsMonday, May 21, 2012
10. 10. Example 1 Find x. a.Monday, May 21, 2012
11. 11. Example 1 Find x. a. m∠FDE = 180 − m∠EDHMonday, May 21, 2012
12. 12. Example 1 Find x. a. m∠FDE = 180 − m∠EDH 76 + 88 m∠EDH = 2Monday, May 21, 2012
13. 13. Example 1 Find x. a. m∠FDE = 180 − m∠EDH 76 + 88 164 m∠EDH = = 2 2Monday, May 21, 2012
14. 14. Example 1 Find x. a. m∠FDE = 180 − m∠EDH 76 + 88 164 m∠EDH = = = 82° 2 2Monday, May 21, 2012
15. 15. Example 1 Find x. a. m∠FDE = 180 − m∠EDH 76 + 88 164 m∠EDH = = = 82° 2 2 m∠FDE = 180 − 82Monday, May 21, 2012
16. 16. Example 1 Find x. a. m∠FDE = 180 − m∠EDH 76 + 88 164 m∠EDH = = = 82° 2 2 m∠FDE = 180 − 82 = 98°Monday, May 21, 2012
17. 17. Example 1 Find x. a. m∠FDE = 180 − m∠EDH 76 + 88 164 m∠EDH = = = 82° 2 2 m∠FDE = 180 − 82 = 98° x = 98Monday, May 21, 2012
18. 18. Example 1 Find x. b.Monday, May 21, 2012
19. 19. Example 1 Find x. b. x = 180 − m∠VZWMonday, May 21, 2012
20. 20. Example 1 Find x. b. x = 180 − m∠VZW 96 + 62 m∠VZW = 2Monday, May 21, 2012
21. 21. Example 1 Find x. b. x = 180 − m∠VZW 96 + 62 m∠VZW = 2 158 = 2Monday, May 21, 2012
22. 22. Example 1 Find x. b. x = 180 − m∠VZW 96 + 62 m∠VZW = 2 158 = = 79° 2Monday, May 21, 2012
23. 23. Example 1 Find x. b. x = 180 − m∠VZW 96 + 62 m∠VZW = 2 158 = = 79° 2 x = 180 − 79Monday, May 21, 2012
24. 24. Example 1 Find x. b. x = 180 − m∠VZW 96 + 62 m∠VZW = 2 158 = = 79° 2 x = 180 − 79 = 101Monday, May 21, 2012
25. 25. Example 1 Find x. c.Monday, May 21, 2012
26. 26. Example 1 Find x. c. x + 25 60 = 2Monday, May 21, 2012
27. 27. Example 1 Find x. c. x + 25 60 = 2 120 = x + 25Monday, May 21, 2012
28. 28. Example 1 Find x. c. x + 25 60 = 2 120 = x + 25 x = 95Monday, May 21, 2012
29. 29. Example 2 Find each measure.  = 250° a. m∠QPS when mPTSMonday, May 21, 2012
30. 30. Example 2 Find each measure.  = 250° a. m∠QPS when mPTS 1  m∠QPS = mPTS 2Monday, May 21, 2012
31. 31. Example 2 Find each measure.  = 250° a. m∠QPS when mPTS 1  m∠QPS = mPTS 2 1 = (250) 2Monday, May 21, 2012
32. 32. Example 2 Find each measure.  = 250° a. m∠QPS when mPTS 1  m∠QPS = mPTS 2 1 = (250) = 125° 2Monday, May 21, 2012
33. 33. Example 2 Find each measure.  b. mBDMonday, May 21, 2012
34. 34. Example 2 Find each measure.  b. mBD  = 360 − 2m∠ADB mBDMonday, May 21, 2012
35. 35. Example 2 Find each measure.  b. mBD  = 360 − 2m∠ADB mBD = 360 − 2(108)Monday, May 21, 2012
36. 36. Example 2 Find each measure.  b. mBD  = 360 − 2m∠ADB mBD = 360 − 2(108) = 360 − 216Monday, May 21, 2012
37. 37. Example 2 Find each measure.  b. mBD  = 360 − 2m∠ADB mBD = 360 − 2(108) = 360 − 216 = 144°Monday, May 21, 2012
38. 38. Example 3 Find each measure.  when m∠AED = 62° a. mBCMonday, May 21, 2012
39. 39. Example 3 Find each measure.  when m∠AED = 62° a. mBC  − mBC mABD  m∠AED = 2Monday, May 21, 2012
40. 40. Example 3 Find each measure.  when m∠AED = 62° a. mBC  − mBC mABD  m∠AED = 2 141 − x 62 = 2Monday, May 21, 2012
41. 41. Example 3 Find each measure.  when m∠AED = 62° a. mBC  − mBC mABD  m∠AED = 2 141 − x 62 = 124 = 141 − x 2Monday, May 21, 2012
42. 42. Example 3 Find each measure.  when m∠AED = 62° a. mBC  − mBC mABD  m∠AED = 2 141 − x 62 = 124 = 141 − x 2 −17 = −xMonday, May 21, 2012
43. 43. Example 3 Find each measure.  when m∠AED = 62° a. mBC  − mBC mABD  m∠AED = 2 141 − x 62 = 124 = 141 − x 2 −17 = −x x = 17Monday, May 21, 2012
44. 44. Example 3 Find each measure.  when m∠AED = 62° a. mBC  − mBC mABD  m∠AED = 2 141 − x 62 = 124 = 141 − x 2 −17 = −x  = 17° mBC x = 17Monday, May 21, 2012
45. 45. Example 3 Find each measure.  b. m XYZMonday, May 21, 2012
46. 46. Example 3 Find each measure.  b. m XYZ  − m XZ m XYZ  m∠W = 2Monday, May 21, 2012
47. 47. Example 3 Find each measure.  b. m XYZ  − m XZ m XYZ  m∠W = 2  − 140 m XYZ 40 = 2Monday, May 21, 2012
48. 48. Example 3 Find each measure.  b. m XYZ  − m XZ m XYZ  m∠W = 2  − 140 m XYZ 40 = 2  − 140 80 = m XYZMonday, May 21, 2012
49. 49. Example 3 Find each measure.  b. m XYZ  − m XZ m XYZ  m∠W = 2  − 140 m XYZ 40 = 2  − 140 80 = m XYZ  = 220° m XYZMonday, May 21, 2012
50. 50. Check Your Understanding p. 731 #1-7Monday, May 21, 2012
51. 51. Problem SetMonday, May 21, 2012
52. 52. Problem Set p. 732 #9-29 odd, 41, 47 "I hate quotations. Tell me what you know." – Ralph Waldo EmersonMonday, May 21, 2012