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# Integrated Math 2 Section 5-2

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Angles and Perpendicular Lines

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### Integrated Math 2 Section 5-2

1. 1. SECTION 5-2 Angles and Perpendicular Lines
2. 2. Essential Questions How do you identify and use perpendicular lines? How do you identify and use angle relationships? Where you’ll see this: Health, physics, paper folding, navigation
3. 3. Vocabulary 1. Ray: 2. Opposite Rays: 3. Angle: 4. Vertex: 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
4. 4. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: 3. Angle: 4. Vertex: 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
5. 5. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: Two rays that together form a straight line 3. Angle: 4. Vertex: 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
6. 6. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: Two rays that together form a straight line 3. Angle: A ﬁgure made up of two rays that share an endpoint 4. Vertex: 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
7. 7. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: Two rays that together form a straight line 3. Angle: A ﬁgure made up of two rays that share an endpoint 4. Vertex: The point that is shared by the rays of an angle 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
8. 8. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: Two rays that together form a straight line 3. Angle: A ﬁgure made up of two rays that share an endpoint 4. Vertex: The point that is shared by the rays of an angle 5. Degrees: The size measurement of an angle 6. Complementary Angles: 7. Supplementary Angles:
9. 9. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: Two rays that together form a straight line 3. Angle: A ﬁgure made up of two rays that share an endpoint 4. Vertex: The point that is shared by the rays of an angle 5. Degrees: The size measurement of an angle 6. Complementary Angles: Two angles that add up to 90 degrees 7. Supplementary Angles:
10. 10. Vocabulary 1. Ray: Part of a line that begins at an endpoint and goes on forever in one direction 2. Opposite Rays: Two rays that together form a straight line 3. Angle: A ﬁgure made up of two rays that share an endpoint 4. Vertex: The point that is shared by the rays of an angle 5. Degrees: The size measurement of an angle 6. Complementary Angles: Two angles that add up to 90 degrees 7. Supplementary Angles: Two angles that add up to 180 degrees
11. 11. Vocabulary 8. Adjacent Angles: 9. Congruent Angles: 10. Perpendicular Lines: 11. Vertical Angles: 12. Bisector of an Angle:
12. 12. Vocabulary 8. Adjacent Angles: Two angles that share a side and vertex but no other points 9. Congruent Angles: 10. Perpendicular Lines: 11. Vertical Angles: 12. Bisector of an Angle:
13. 13. Vocabulary 8. Adjacent Angles: Two angles that share a side and vertex but no other points 9. Congruent Angles: Two or more angles with the same measure 10. Perpendicular Lines: 11. Vertical Angles: 12. Bisector of an Angle:
14. 14. Vocabulary 8. Adjacent Angles: Two angles that share a side and vertex but no other points 9. Congruent Angles: Two or more angles with the same measure 10. Perpendicular Lines: Lines that intersect at a 90 degree angle 11. Vertical Angles: 12. Bisector of an Angle:
15. 15. Vocabulary 8. Adjacent Angles: Two angles that share a side and vertex but no other points 9. Congruent Angles: Two or more angles with the same measure 10. Perpendicular Lines: Lines that intersect at a 90 degree angle 11. Vertical Angles: When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent 12. Bisector of an Angle:
16. 16. Vocabulary 8. Adjacent Angles: Two angles that share a side and vertex but no other points 9. Congruent Angles: Two or more angles with the same measure 10. Perpendicular Lines: Lines that intersect at a 90 degree angle 11. Vertical Angles: When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent 12. Bisector of an Angle: A ray that divides an angle into two equal angles
17. 17. Ray Part of a line that begins at an endpoint and goes on forever in one direction
18. 18. Ray Part of a line that begins at an endpoint and goes on forever in one direction A B
19. 19. Opposite Rays Two rays that together form a straight line
20. 20. Opposite Rays Two rays that together form a straight line A B
21. 21. Opposite Rays Two rays that together form a straight line C A B
22. 22. Angle A ﬁgure made up of two rays that share an endpoint
23. 23. Angle A ﬁgure made up of two rays that share an endpoint C A B
24. 24. Vertex The point that is shared by the rays of an angle C A B
25. 25. Vertex The point that is shared by the rays of an angle C A B
26. 26. Degrees The size measurement of an angle C A B
27. 27. Degrees The size measurement of an angle C A B http://www.chutedesign.co.uk/design/protractor/protractor.gif
28. 28. Complementary Angles Two angles that add up to 90 degrees
29. 29. Complementary Angles Two angles that add up to 90 degrees D A C B
30. 30. Supplementary Angles Two angles that add up to 180 degrees
31. 31. Supplementary Angles Two angles that add up to 180 degrees C D B A
32. 32. Adjacent Angles Two angles that share a side and vertex but no other points C D B A
33. 33. Congruent Angles Two or more angles with the same measure
34. 34. Congruent Angles Two or more angles with the same measure A M N
35. 35. Perpendicular Lines Lines that intersect at a 90 degree angle
36. 36. Perpendicular Lines Lines that intersect at a 90 degree angle g h
37. 37. Perpendicular Lines Lines that intersect at a 90 degree angle g r h s
38. 38. Perpendicular Lines Lines that intersect at a 90 degree angle g r h s perpendicular
39. 39. Perpendicular Lines Lines that intersect at a 90 degree angle g r h s perpendicular NOT perpendicular
40. 40. Vertical Angles When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent
41. 41. Vertical Angles When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent B A E C D
42. 42. Vertical Angles When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent B A E C D
43. 43. Vertical Angles When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent B A E C D
44. 44. Bisector of an Angle A ray that divides an angle into two equal angles
45. 45. Bisector of an Angle A ray that divides an angle into two equal angles N A D
46. 46. Bisector of an Angle A ray that divides an angle into two equal angles N E A D
47. 47. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle.
48. 48. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle:
49. 49. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: Larger angle:
50. 50. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle:
51. 51. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40
52. 52. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180
53. 53. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 2x + 40 =180
54. 54. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 2x + 40 =180 −40 −40
55. 55. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 2x + 40 =180 −40 −40 2x =140
56. 56. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 2x + 40 =180 −40 −40 2x =140 2 2
57. 57. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 2x + 40 =180 −40 −40 2x =140 2 2 x = 70
58. 58. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 70 + 40 2x + 40 =180 −40 −40 2x =140 2 2 x = 70
59. 59. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 70 + 40 =110 2x + 40 =180 −40 −40 2x =140 2 2 x = 70
60. 60. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle: x Larger angle: x + 40 x + x + 40 =180 70 + 40 =110 2x + 40 =180 −40 −40 2x =140 The angles are 70° and 110° 2 2 x = 70
61. 61. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD.
62. 62. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠AQE = 4 0° E F Q C D
63. 63. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° m∠AQE = 4 0° E F Q C D
64. 64. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° E F Q C D
65. 65. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° E F Q C D
66. 66. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° E F Q C D
67. 67. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E F Q C D
68. 68. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E F Q C D
69. 69. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E m∠BQF = 40° F Q C D
70. 70. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E m∠BQF = 40° (vertical angles) F Q C D
71. 71. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E m∠BQF = 40° (vertical angles) F Q C D
72. 72. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E m∠BQF = 40° (vertical angles) F Q m∠BQD = m∠BQF + m∠FQD C D
73. 73. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E m∠BQF = 40° (vertical angles) F Q m∠BQD = m∠BQF + m∠FQD C = 40° + 40° = 80° D
74. 74. Example 2 Draw the ﬁgure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD. A B m∠FQD = 40° (vertical angles) m∠AQE = 4 0° m∠EQC = 40° (angle bisector) E m∠BQF = 40° (vertical angles) F Q m∠BQD = m∠BQF + m∠FQD C = 40° + 40° = 80° D m∠BQD = 80°
75. 75. Example 3 If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C supplementary angles? Explain.
76. 76. Example 3 If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C supplementary angles? Explain. 20° + 60° +100° =180°?
77. 77. Example 3 If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C supplementary angles? Explain. 20° + 60° +100° =180°? Yes. Since the sum of the three angles is 180°, they are supplementary.
78. 78. Example 4 Are angles 1 and 2 adjacent? Explain. 1 2
79. 79. Example 4 Are angles 1 and 2 adjacent? Explain. 1 2 These are not adjacent. In order to be adjacent, the two angles must share both the vertex and a side.
80. 80. Example 5 Are ∠AEB and∠CED vertical angles? Explain. B A E C D
81. 81. Example 5 Are ∠AEB and∠CED vertical angles? Explain. B A E C D There are no vertical angles here, as none of the angles are formed by segments intersecting any any place other than the endpoints
82. 82. Homework
83. 83. Homework p. 198 #1-24 “No problem is too small or too trivial if we can really do something about it.” - Richard Feynman