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Chapter 3

Chapter 3 . Parallel and Perpendicular Lines. 3.1 Properties of Parallel Lines. Transversal – a line that intersects two coplanar lines at two distinct points. Transversal. Types of Angles. Exterior (The outside of the two lines). Interior (The inside of the two lines).

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Chapter 3

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  1. Chapter 3 Parallel and Perpendicular Lines

  2. 3.1 Properties of Parallel Lines Transversal– a line that intersects two coplanar lines at two distinct points Transversal

  3. Types of Angles Exterior (The outside of the two lines) Interior (The inside of the two lines)

  4. Types of Angles Alternate Interior Angles B A C D On alternate sides of the transversal and inside the two lines

  5. Types of Angles Same-Side Interior Angles B A C D On the same side of the transversal and inside the two lines

  6. Types of Angles Alternate Exterior Angles B A C D On alternate sides of the transversal and outside the two lines

  7. Types of Angles Same-Side Exterior Angles B A C D On the same side of the transversal and outside the two lines

  8. Types of Angles Corresponding Angles B A D C E F G H Angles in the corresponding position of the transversal

  9. Solve for each angle.

  10. Conclusions for Parallel Lines: Corresponding Angles are ___________ Alternate Interior Angles are ___________ Same-Side Interior Angles are ___________ Alternate Exterior Angles are ___________ Same-Side Exterior Angles are ___________

  11. Corresponding Angles Postulate: If a transversal intersects two parallel lines, then corresponding angles are congruent.

  12. Alternate Interior Angles Theorem: If a transversal intersects two parallel lines, then alternate interior angles are congruent.

  13. Same-Side Interior Angles Theorem: If a transversal intersects two parallel lines, then same side interior angles are supplementary.

  14. Alternate Exterior Angles Theorem: If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

  15. Same-Side Exterior Angles Theorem: If a transversal intersects two parallel lines, then same side exterior angles are supplementary.

  16. Mixed Practice Find the measure of all of the numbered angles in the following diagram. 115°

  17. Checkpoint Find the value of x. Use Algebra with Angle Relationships 16. 85 ANSWER 17. 104 ANSWER 18. 40 ANSWER

  18. Example Proof: Given: Prove:

  19. The Converses: used to prove lines parallel Converse of the Corresponding Angles Postulate: If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. Notice: The angle relationship must be true before you can say the lines are parallel

  20. Converse of Alternate Interior Angles Theorem: If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. 56o 56o

  21. Converse of Same-Side Interior Angles Theorem: If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. 56o 124o

  22. Converse of Alternate Exterior Angles Theorem: If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. 56o 56o

  23. Converse of Same-Side Exterior Angles Theorem: If two lines and a transversal form same-side exterior angles that are supplementary, then the two lines are parallel. 56o 124o

  24. Find the value of x that makes m || n.

  25. Find the value of x that makes m || n and r || s.

  26. Given: m || n Prove: r || p Given Given Corr Angles Postulate Transitive Converse Alt Int Theorem

  27. Given: m || n Prove: r || p Given Given Corr Angles Postulate Transitive Converse Alt Int Theorem

  28. Warm Up: Given: a || b c || b Prove: a || c

  29. 3.3 Parallel and Perpendicular Lines

  30. Theorem 3-9 If two lines are parallel to the same line, then they are parallel to each other. If m || n and n || p then m || p.

  31. Given: Prove:

  32. Theorem 3-10 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

  33. Given: Prove:

  34. Theorem 3-11 In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

  35. Theorem 3-9 If two lines are parallel to the same line, then they are parallel to each other. n m p If n || m and m||p then n||p.

  36. Theorem 3-10: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. a b c

  37. Given: Prove: l 1 m Do not use the transitive property! 2 n 3 q

  38. Given: Prove: a 1 b Do not substitute 90˚ for right! 2 c 3 d

  39. Theorem 3-11 In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. j h k

  40. Determine the relationship, if any, of lines a and e.

  41. 3.4 Parallel Lines and the Triangle Angle-Sum Theorem

  42. Triangles by Angles: Acute – No angles larger than 90 Equiangular – all angles are congruent Obtuse – one angle larger than 90 Right – One right angle

  43. Triangles by Sides Scalene– No congruent sides Isosceles – At least 2 congruent sides Equilateral – All sides are congruent

  44. Triangle Angle-Sum Theorem A The sum of the measures of the angles of a triangle is 180˚ B C

  45. Triangle Angle Sum Theorem The interior angles of a triangle sum to 180º. Solve for x:

  46. Angles of a polygon: Exterior Angle Remote Interior Angles The two nonadjacent interior angles to an exterior angle

  47. Triangle Exterior Angle Theorem: The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

  48. Example: Find the value of x. 1) 2)

  49. Find the values of the variables:

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