Integrated Math 2 Section 5-2
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Integrated Math 2 Section 5-2

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Angles and Perpendicular Lines

Angles and Perpendicular Lines

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Integrated Math 2 Section 5-2 Integrated Math 2 Section 5-2 Presentation Transcript

  • 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 any any place other than the endpoints
  • Homework
  • Homework p. 198 #1-24 “No problem is too small or too trivial if we can really do something about it.” - Richard Feynman