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# 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