Geometry Section 10-4 1112

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Inscribed Angles

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Geometry Section 10-4 1112

  1. 1. Section 10-4 Inscribed AnglesThursday, May 17, 2012
  2. 2. Essential Questions How do you find measures of inscribed angles? How do you find measures of angles on inscribed polygons?Thursday, May 17, 2012
  3. 3. Vocabulary 1. Inscribed Angle: 2. Intercepted Arc:Thursday, May 17, 2012
  4. 4. Vocabulary 1. Inscribed Angle: An angle made of two chords in a circle, so that the vertex is on the edge of the circle 2. Intercepted Arc:Thursday, May 17, 2012
  5. 5. Vocabulary 1. Inscribed Angle: An angle made of two chords in a circle, so that the vertex is on the edge of the circle 2. Intercepted Arc: An arc with endpoints on the sides of an inscribed angle and in the interior of the inscribed angleThursday, May 17, 2012
  6. 6. Theorems 10.6 - Inscribed Angle Theorem: 10.7 - Two Inscribed Angles: 10.8 - Inscribed Angles and Diameters:Thursday, May 17, 2012
  7. 7. Theorems 10.6 - Inscribed Angle Theorem: If an angle is inscribed in a circle, then the measure of the angle is one half the measure of the intercepted arc 10.7 - Two Inscribed Angles: 10.8 - Inscribed Angles and Diameters:Thursday, May 17, 2012
  8. 8. Theorems 10.6 - Inscribed Angle Theorem: If an angle is inscribed in a circle, then the measure of the angle is one half the measure of the intercepted arc 10.7 - Two Inscribed Angles: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent 10.8 - Inscribed Angles and Diameters:Thursday, May 17, 2012
  9. 9. Theorems 10.6 - Inscribed Angle Theorem: If an angle is inscribed in a circle, then the measure of the angle is one half the measure of the intercepted arc 10.7 - Two Inscribed Angles: If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent 10.8 - Inscribed Angles and Diameters: An inscribed angle of a triangle intercepts a diameter or semicircle IFF the angle is a right angleThursday, May 17, 2012
  10. 10. Example 1 Find each measure. a. m∠YXW  b. m XZThursday, May 17, 2012
  11. 11. Example 1 Find each measure. a. m∠YXW 1  m∠YXW = mYW 2  b. m XZThursday, May 17, 2012
  12. 12. Example 1 Find each measure. a. m∠YXW 1  1 m∠YXW = mYW = (86) 2 2  b. m XZThursday, May 17, 2012
  13. 13. Example 1 Find each measure. a. m∠YXW 1  1 m∠YXW = mYW = (86) = 43° 2 2  b. m XZThursday, May 17, 2012
  14. 14. Example 1 Find each measure. a. m∠YXW 1  1 m∠YXW = mYW = (86) = 43° 2 2  b. m XZ  m XZ = 2m∠XYZThursday, May 17, 2012
  15. 15. Example 1 Find each measure. a. m∠YXW 1  1 m∠YXW = mYW = (86) = 43° 2 2  b. m XZ  m XZ = 2m∠XYZ = 2(52)Thursday, May 17, 2012
  16. 16. Example 1 Find each measure. a. m∠YXW 1  1 m∠YXW = mYW = (86) = 43° 2 2  b. m XZ  m XZ = 2m∠XYZ = 2(52) =104°Thursday, May 17, 2012
  17. 17. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°.Thursday, May 17, 2012
  18. 18. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2Thursday, May 17, 2012
  19. 19. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2 3x =15Thursday, May 17, 2012
  20. 20. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2 3x =15 x =5Thursday, May 17, 2012
  21. 21. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2 3x =15 x =5 m∠QRT =12(5)−13Thursday, May 17, 2012
  22. 22. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2 3x =15 x =5 m∠QRT =12(5)−13 = 60 −13Thursday, May 17, 2012
  23. 23. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2 3x =15 x =5 m∠QRT =12(5)−13 = 60 −13 = 47°Thursday, May 17, 2012
  24. 24. Example 3 Prove the following.   Given: LO ≅ MN Prove: MNP ≅LOPThursday, May 17, 2012
  25. 25. Example 3 Prove the following.   Given: LO ≅ MN Prove: MNP ≅LOP There are many ways to prove this one. Work through a proof on your own. We will discuss a few in class.Thursday, May 17, 2012
  26. 26. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°.Thursday, May 17, 2012
  27. 27. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180Thursday, May 17, 2012
  28. 28. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180Thursday, May 17, 2012
  29. 29. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180 9x + 90 =180Thursday, May 17, 2012
  30. 30. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180 9x + 90 =180 9x = 90Thursday, May 17, 2012
  31. 31. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180 9x + 90 =180 9x = 90 x =10Thursday, May 17, 2012
  32. 32. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180 9x + 90 =180 9x = 90 x =10 m∠B = 8(10)− 4Thursday, May 17, 2012
  33. 33. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180 9x + 90 =180 9x = 90 x =10 m∠B = 8(10)− 4 = 80 − 4Thursday, May 17, 2012
  34. 34. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°. m∠A + m∠B + m∠C =180 x + 4 + 8x − 4 + 90 =180 9x + 90 =180 9x = 90 x =10 m∠B = 8(10)− 4 = 80 − 4 = 76°Thursday, May 17, 2012
  35. 35. Check Your Understanding p. 713 #1-10Thursday, May 17, 2012
  36. 36. Problem SetThursday, May 17, 2012
  37. 37. Problem Set p. 713 #11-35 odd, 49, 55, 61 “Youre alive. Do something. The directive in life, the moral imperative was so uncomplicated. It could be expressed in single words, not complete sentences. It sounded like this: Look. Listen. Choose. Act.” - Barbara HallThursday, May 17, 2012

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