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# Int Math 2 Section 5-8 1011

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Properties of Circles

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### Int Math 2 Section 5-8 1011

1. 1. Section 5-8 Properties of CirclesWed, Feb 02
2. 2. Essential Questions • What are the relationships among parts of a circle? • What are the properties of circles and how do you apply them? • Where you’ll see this: • Market research, food service, art, recreation, navigationWed, Feb 02
3. 3. Vocabulary 1. Circle: 2. Radius: 3. Chord: 4. Diameter: 5. Central Angle:Wed, Feb 02
4. 4. Vocabulary 1. Circle: All points that are the same distance from a ﬁxed center point; 360° total 2. Radius: 3. Chord: 4. Diameter: 5. Central Angle:Wed, Feb 02
5. 5. Vocabulary 1. Circle: All points that are the same distance from a ﬁxed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: 4. Diameter: 5. Central Angle:Wed, Feb 02
6. 6. Vocabulary 1. Circle: All points that are the same distance from a ﬁxed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: A segment where both endpoints are on the circle 4. Diameter: 5. Central Angle:Wed, Feb 02
7. 7. Vocabulary 1. Circle: All points that are the same distance from a ﬁxed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: A segment where both endpoints are on the circle 4. Diameter: A chord that goes through the center of a circle 5. Central Angle:Wed, Feb 02
8. 8. Vocabulary 1. Circle: All points that are the same distance from a ﬁxed center point; 360° total 2. Radius: A segment whose endpoints are the center of a circle and on the circle 3. Chord: A segment where both endpoints are on the circle 4. Diameter: A chord that goes through the center of a circle 5. Central Angle: An angle where the vertex is the center of the circleWed, Feb 02
9. 9. Vocabulary 6. Arc: 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:Wed, Feb 02
10. 10. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:Wed, Feb 02
11. 11. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: 9. Major Arc: 10. Inscribed Angle:Wed, Feb 02
12. 12. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: An arc that is less than half the circumference; same measure as the central angle 9. Major Arc: 10. Inscribed Angle:Wed, Feb 02
13. 13. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: An arc that is less than half the circumference; same measure as the central angle 9. Major Arc: An arc that is more than half the circumference 10. Inscribed Angle:Wed, Feb 02
14. 14. Vocabulary 6. Arc: A section of the circumference of a circle 7. Semicircle: An arc that is half of the circumference; half a circle 8. Minor Arc: An arc that is less than half the circumference; same measure as the central angle 9. Major Arc: An arc that is more than half the circumference 10. Inscribed Angle: An angle whose vertex is on the circle and whose sides are chords of the circle; half the measure of the arc it containsWed, Feb 02
15. 15. CircleWed, Feb 02
17. 17. ChordWed, Feb 02
18. 18. DiameterWed, Feb 02
19. 19. Central AngleWed, Feb 02
20. 20. ArcWed, Feb 02
21. 21. SemicircleWed, Feb 02
22. 22. Minor ArcWed, Feb 02
23. 23. Major ArcWed, Feb 02
24. 24. Inscribed AngleWed, Feb 02
25. 25. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB Wed, Feb 02
26. 26. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132°Wed, Feb 02
27. 27. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82°Wed, Feb 02
28. 28. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° x°Wed, Feb 02
29. 29. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° x° x°Wed, Feb 02
30. 30. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° x° x°Wed, Feb 02
31. 31. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 x° x°Wed, Feb 02
32. 32. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 x° x°Wed, Feb 02
33. 33. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 x° x°Wed, Feb 02
34. 34. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 x° x°Wed, Feb 02
35. 35. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 x° x° x = 73Wed, Feb 02
36. 36. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  x + x +132 + 82 = 360 132° 82° 2x + 214 = 360 −214 −214 2x =146 2 2 73° 73° x = 73Wed, Feb 02
37. 37. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
38. 38. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 73° 73°Wed, Feb 02
39. 39. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 1 = (73 + 73) 2 73° 73°Wed, Feb 02
40. 40. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 1 1 = (73 + 73) = (146) 2 2 73° 73°Wed, Feb 02
41. 41. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠ABC = (mAD + mCD 2 1 1 = (73 + 73) = (146) = 73° 2 2 73° 73°Wed, Feb 02
42. 42. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
43. 43. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 73° 73°Wed, Feb 02
44. 44. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 1 = (73 +132) 2 73° 73°Wed, Feb 02
45. 45. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 1 1 = (73 +132) = (205) 2 2 73° 73°Wed, Feb 02
46. 46. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠BCD = (mAD + mAB 2 1 1 = (73 +132) = (205) =102.5° 2 2 73° 73°Wed, Feb 02
47. 47. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
48. 48. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 73° 73°Wed, Feb 02
49. 49. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 1 = (82 +132) 2 73° 73°Wed, Feb 02
50. 50. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 1 1 = (82 +132) = (214) 2 2 73° 73°Wed, Feb 02
51. 51. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠CDA = (mBC + mAB 2 1 1 = (82 +132) = (214) =107° 2 2 73° 73°Wed, Feb 02
52. 52. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° 73° 73°Wed, Feb 02
53. 53. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 73° 73°Wed, Feb 02
54. 54. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 1 = (82 + 73) 2 73° 73°Wed, Feb 02
55. 55. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 1 1 = (82 + 73) = (155) 2 2 73° 73°Wed, Feb 02
56. 56. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  1  ) 132° 82° m∠DAB = (mBC + mCD 2 1 1 = (82 + 73) = (155) = 77.5° 2 2 73° 73°Wed, Feb 02
57. 57. Example 1  ≅ CD . Find the measures of the  In circle O, AD angles of quadrilateral ABCD, when  =132° and mBC = 82°. mAB  132° 82° m∠ABC = 73° m∠BCD =102.5° m∠CDA =107° 73° 73° m∠DAB = 77.5°Wed, Feb 02
58. 58. Example 2 Identify the following for circle P. a. Diameter b. Radius c. Chord  d. mLM   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
59. 59. Example 2 Identify the following for circle P. a. Diameter b. Radius JK c. Chord  d. mLM   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
60. 60. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
61. 61. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
62. 62. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47°   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
63. 63. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ g. m∠LKJ h. Central AngleWed, Feb 02
64. 64. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° g. m∠LKJ h. Central AngleWed, Feb 02
65. 65. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° g. m∠LKJ h. Central AngleWed, Feb 02
66. 66. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central AngleWed, Feb 02
67. 67. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) 1Wed, Feb 02
68. 68. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) = 31° 1Wed, Feb 02
69. 69. Example 2 Identify the following for circle P. a. Diameter b. Radius JK KP c. Chord  d. mLM KL = 62° + 47° =109°   e. mLMK f. mLJ = 62° +180° = 242° = 62° g. m∠LKJ h. Central Angle = 2 (62°) = 31° 1 ∠JPMWed, Feb 02
70. 70. Problem SetWed, Feb 02
71. 71. Problem Set p. 228 #1-25 odd “We are so accustomed to disguise ourselves to others that in the end we become disguised to ourselves.” - Francois de La RochefoucauldWed, Feb 02