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Int Math 2 Section 5-2 1011

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Jimbo Lamb

Let x = the measure of the smaller angle Then the measure of the larger angle is x + 40 We know: x + (x + 40) = 180 2x + 40 = 180 2x = 140 x = 70 Therefore, the measures of the angles are: Smaller angle: 70° Larger angle: 70° + 40° = 110°

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SECTION 5-2 Angles and Perpendicular Lines
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
Vocabulary 1. Ray: 2. Opposite Rays: 3. Angle: 4. Vertex: 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
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:
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:
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:
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:
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:
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:
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
Vocabulary 8. Adjacent Angles: 9. Congruent Angles: 10. Perpendicular Lines: 11. Vertical Angles: 12. Bisector of an Angle:
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:
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:
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:
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:
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
Ray Part of a line that begins at an endpoint and goes on forever in one direction
Ray Part of a line that begins at an endpoint and goes on forever in one direction A B
Opposite Rays Two rays that together form a straight line
Opposite Rays Two rays that together form a straight line A B
Opposite Rays Two rays that together form a straight line C A B
Angle A figure made up of two rays that share an endpoint
Angle A figure made up of two rays that share an endpoint C A B
Vertex The point that is shared by the rays of an angle C A B
Vertex The point that is shared by the rays of an angle C A B
Degrees The size measurement of an angle C A B
Degrees The size measurement of an angle C A B http://www.chutedesign.co.uk/design/protractor/protractor.gif
Complementary Angles Two angles that add up to 90 degrees
Complementary Angles Two angles that add up to 90 degrees D A C B
Supplementary Angles Two angles that add up to 180 degrees
Supplementary Angles Two angles that add up to 180 degrees C D B A
Adjacent Angles Two angles that share a side and vertex but no other points C D B A
Congruent Angles Two or more angles with the same measure
Congruent Angles Two or more angles with the same measure A M N
Perpendicular Lines Lines that intersect at a 90 degree angle
Perpendicular Lines Lines that intersect at a 90 degree angle g h
Perpendicular Lines Lines that intersect at a 90 degree angle g r h s
Perpendicular Lines Lines that intersect at a 90 degree angle g r h s perpendicular
Perpendicular Lines Lines that intersect at a 90 degree angle g r h s perpendicular NOT perpendicular
Vertical Angles When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent
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
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
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
Bisector of an Angle A ray that divides an angle into two equal angles
Bisector of an Angle A ray that divides an angle into two equal angles N A D
Bisector of an Angle A ray that divides an angle into two equal angles N E A D
Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle.
Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle. Smaller angle:
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:
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:
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
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
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
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
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
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
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
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
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
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
Example 2 Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD.
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
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
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
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
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
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
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
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
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
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
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
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
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°
Example 3 If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C supplementary angles? Explain.
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°?
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.
Example 4 Are angles 1 and 2 adjacent? Explain. 1 2
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.
Example 5 Are ∠AEB and∠CED vertical angles? Explain. B A E C D
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 at any place other than the endpoints
Problem Set
Problem Set p. 198 #1-24 “No problem is too small or too trivial if we can really do something about it.” - Richard Feynman

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Int Math 2 Section 5-2 1011

  • 1. SECTION 5-2 Angles and Perpendicular Lines
  • 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. Vocabulary 1. Ray: 2. Opposite Rays: 3. Angle: 4. Vertex: 5. Degrees: 6. Complementary Angles: 7. Supplementary Angles:
  • 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. 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. 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. 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. 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. 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. 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. Vocabulary 8. Adjacent Angles: 9. Congruent Angles: 10. Perpendicular Lines: 11. Vertical Angles: 12. Bisector of an Angle:
  • 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. 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. 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. 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. 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. Ray Part of a line that begins at an endpoint and goes on forever in one direction
  • 18. Ray Part of a line that begins at an endpoint and goes on forever in one direction A B
  • 19. Opposite Rays Two rays that together form a straight line
  • 20. Opposite Rays Two rays that together form a straight line A B
  • 21. Opposite Rays Two rays that together form a straight line C A B
  • 22. Angle A figure made up of two rays that share an endpoint
  • 23. Angle A figure made up of two rays that share an endpoint C A B
  • 24. Vertex The point that is shared by the rays of an angle C A B
  • 25. Vertex The point that is shared by the rays of an angle C A B
  • 26. Degrees The size measurement of an angle C A B
  • 27. Degrees The size measurement of an angle C A B http://www.chutedesign.co.uk/design/protractor/protractor.gif
  • 28. Complementary Angles Two angles that add up to 90 degrees
  • 29. Complementary Angles Two angles that add up to 90 degrees D A C B
  • 30. Supplementary Angles Two angles that add up to 180 degrees
  • 31. Supplementary Angles Two angles that add up to 180 degrees C D B A
  • 32. Adjacent Angles Two angles that share a side and vertex but no other points C D B A
  • 33. Congruent Angles Two or more angles with the same measure
  • 34. Congruent Angles Two or more angles with the same measure A M N
  • 35. Perpendicular Lines Lines that intersect at a 90 degree angle
  • 36. Perpendicular Lines Lines that intersect at a 90 degree angle g h
  • 37. Perpendicular Lines Lines that intersect at a 90 degree angle g r h s
  • 38. Perpendicular Lines Lines that intersect at a 90 degree angle g r h s perpendicular
  • 39. Perpendicular Lines Lines that intersect at a 90 degree angle g r h s perpendicular NOT perpendicular
  • 40. Vertical Angles When two lines/segments intersect, the two angles that are not adjacent to each other; they are congruent
  • 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. 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. 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. Bisector of an Angle A ray that divides an angle into two equal angles
  • 45. Bisector of an Angle A ray that divides an angle into two equal angles N A D
  • 46. Bisector of an Angle A ray that divides an angle into two equal angles N E A D
  • 47. Example 1 The larger of the two supplementary angles measures 40° more than the smaller. Find the measure of each angle.
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 2 Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC. m∠AQE = 40°. Find m∠BQD.
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 3 If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C supplementary angles? Explain.
  • 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. 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. Example 4 Are angles 1 and 2 adjacent? Explain. 1 2
  • 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. Example 5 Are ∠AEB and∠CED vertical angles? Explain. B A E C D
  • 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 at any place other than the endpoints
  • 83. Problem Set p. 198 #1-24 “No problem is too small or too trivial if we can really do something about it.” - Richard Feynman

Editor's Notes