Geometry Section 10-4 1112

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Inscribed Angles

Inscribed Angles

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  • 1. Section 10-4 Inscribed AnglesThursday, May 17, 2012
  • 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. Vocabulary 1. Inscribed Angle: 2. Intercepted Arc:Thursday, May 17, 2012
  • 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. 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. Theorems 10.6 - Inscribed Angle Theorem: 10.7 - Two Inscribed Angles: 10.8 - Inscribed Angles and Diameters:Thursday, May 17, 2012
  • 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. 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. 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. Example 1 Find each measure. a. m∠YXW  b. m XZThursday, May 17, 2012
  • 11. Example 1 Find each measure. a. m∠YXW 1  m∠YXW = mYW 2  b. m XZThursday, May 17, 2012
  • 12. Example 1 Find each measure. a. m∠YXW 1  1 m∠YXW = mYW = (86) 2 2  b. m XZThursday, May 17, 2012
  • 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. 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. 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. 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. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°.Thursday, May 17, 2012
  • 18. Example 2 Find m∠QRT when m∠QRT = (12x − 13)° and m∠QST = (9x + 2)°. 12x −13 = 9x + 2Thursday, May 17, 2012
  • 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. 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. 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. 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. 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. Example 3 Prove the following.   Given: LO ≅ MN Prove: MNP ≅LOPThursday, May 17, 2012
  • 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. Example 4 Find m∠B when m∠A = (x + 4)° and m∠B = (8x - 4)°.Thursday, May 17, 2012
  • 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. 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. 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. 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. 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. 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. 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. 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. Check Your Understanding p. 713 #1-10Thursday, May 17, 2012
  • 36. Problem SetThursday, May 17, 2012
  • 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