1. Ch 3 Goals and Common Core Standards Ms. Helgeson
Identify relationships between lines.
Identify angles formed by transversals.
Write different types of proofs.
Prove results about perpendicular lines.
Prove and use results about parallel lines and transversals.

2. Use properties of parallel lines to solve real-life problems.
Prove that two lines are parallel.
Use properties of parallel lines in real-life situations.
Construct parallel lines using straightedge and compass.
Find slopes of lines and use slope to identify parallel lines in a coordinate plane.

3. Write equations of parallel lines in a coordinate plane.
Use slope to identify perpendicular lines in a coordinate plane.
Write equations of perpendicular lines.
CC.9-12.G.CO.9
CC.9-12.G.CO.12

4. CC.9-12.G.GPE.6
CC.9-12.G.GPE.5

5. Parallel and Perpendicular Lines
Topic 2
Parallel and Perpendicular Lines

6. Common Core Standards
CC.9-12.G.CO.9 Students will learn how to prove theorems about lines and angles. Students will learn to identify relationships between lines and angles formed by transversals.

7. Chapter 3 Section 1
Parallel Lines: Two lines that are coplanar and do not intersect.
Skew Lines: Two lines that are not coplanar and do not intersect.
Parallel Planes: Two planes that do not intersect.

8. 3.3 Parallel Lines and Transversals
CC.9-12.F.CO.9 Students will be able to prove theorems about lines and angles. Students will learn to prove and use results about parallel lines and transversals. Students will learn to use properties of parallel lines to solve real-life problems.

9. Corresponding Angles: Two angles that occupy corresponding positions.

10. Alternate Interior Angles: Two angles that lie between the two lines on the different sides of the transversal.

11. Alternate Exterior Angles: Two angles that lie outside the two lines on opposite sides of the transversal.

12. Consecutive Interior Angles (Same Side Interior Angles): Two angles that lie between the two lines on the same side of the transversal.

13. List all pairs of angles that fit the description.
Corresponding
Alternate exterior
Alternate interior
Consecutive interior

14. <9 and <11 are _______ angles.
Complete the statement using corresponding, alternate exterior, alternate interior, or consecutive interior.
<9 and <11 are _______ angles.
<6 and <10 are _______ angles.
<8 and <11 are _______ angles.
<7 and <13 are _______ angles. 6
8 7 10

15. Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2
<1=<2

16. Theorem 3.4 Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4
<3 =<4

17. Prove Alternate Interior Angles are Congruent
Given: p ║ q Prove: < 1 ≅ < 2 *Answer on next slide* p 1 2 q 3

18. < 1 ≅ < 3 2. Corr. Angle Pos.
p ║ q given
< 1 ≅ < Corr. Angle Pos.
< 2 ≅ < Vertical Angle Thm
< 1 ≅ < Trans. Prop. of ≅ p 1 2 q 3

19. Theorem Consecutive Interior Angles Theorem (workbook- same side interior angles)
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 6
m<5+m<6 = 180º

20. Prove the Consecutive Interior Angle Theorem
Given: p ║ q Prove: < 1 and < 2 are supp. *Answer on next slide* p 1 2 q 3

21. < 1 ≅ < 3 2. Corr. Angle Pos.
G: p ║ q
P. < 1 and < 2 are supp.
p ║ q Given
< 1 ≅ < Corr. Angle Pos.
m<1 = m< Def. of ≅ <‘s
< 3 and < 2 are Def of linear pair
a linear pair
5. <3 & < 2 are supp linear pair pos.
6. m<3+m<2 = Def. of supp. <‘s
7. m<1 + m<2 = Subst. P. of =
8. < 1 and < 2 are supp 8. Def. of supp. 1 2 q 3

22. Theorem 3.6 Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8
<7 =<8

23. Theorem 3.1
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
Theorem 3.2
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem 3.3
If two line are perpendicular, then they intersect to form four right angles.
g h g h

24. Theorem 3.7 Perpendicular Transversal Theorem
If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. j h k
j k

25. Given that m<1= 118º, find each measure.
<2
<3
<4
<5 1 2 3 5 4

26. Use properties of parallel lines to find the value of x.
39º 1

27. Use properties of parallel lines to find the value of x.

28. Use properties of parallel lines to find the value of x.

30. 2.2 Proving Lines are Parallel
Students will learn to prove that two lines are parallel and use properties of parallel lines to solve real life problems.

31. Chapter 3 Section 4 Corresponding Angles Converse Postulate
If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. j 6 k 1
If <1 = <6, then j k.

32. Theorem 3.8 Alternate Interior Angles Converse
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. j 3 1 k
If <1 = <3, then j k.

33. Given: < 1 ≅ < 2 Prove: j ║ k 3 j 2 1 k

34. Given: < 1 ≅ < 2 Prove: j ║ k 1. < 1 ≅ < 2 1. Given 2
Given: < 1 ≅ < 2 Prove: j ║ k 1. < 1 ≅ < 2 1. Given 2. < 2 ≅ < 3 2. Vert. Angle Thm. 3. < 1 ≅ < 3 3. Trans. Prop of ≅ 4. J ║ k 4. Corr. Angle converse 3 j 2 1 k

35. Theorem 3.9 Consecutive Interior Angles Converse
If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. j 2 1 k
If m<1 + m<2 = 180º, then j k.

36. Given: < 1 & < 2 are Supp Prove: j ║ k *Answer on next page* 3 q

37. Given: <1 & <2 are supp Prove: j ║ k
<1 & <2 are supp Given
<2 & <3 are linear pair Def. of lin. Pair
<2 & < 3 are supp linear pair pos.
< 1 ≅ < Cong. Supp. Pos
j ║ k Alt. Int. Angle
Converse j 1 3 2 k

38. Theorem 3.10 Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. 4 j k 5
If <4 = <5, then j k.

39. Find the value of x that makes m n.

40. Find the value of x that makes p q.

41. 1) Is AB ║DC?
2) Is BC ║AD?
155 D
115 C 65 40 65 B A

42. Answers:
Yes No

43. Given: m<7 = 125, m<8 = 55
Prove: j k j 7 8 k

44. Given: a b, <1 <2
Prove: c d c d a 1 2 3 b

45. 2.< 1 & < 3 are supp 2) Consec. Interior < Thm d
G: a ║ b; < 1 ≅ < 2
P: c ║ d
a ║ b Given
2.< 1 & < 3 are supp 2) Consec. Interior < Thm
3. m<1+m<3= Def of supp. Angles
4. < 1 ≅ < Given
5. m<1 = m< Def. of congruent angles
6. m<2+m<3= Subst. Prop. of equality
7. < 2 & < 3 are supp Def. of supp. angles
8. c ║ d Consecutive Interior
angle converse a 1 2 3 b

46. Theorem: If two lines are parallel to the same line, then they are parallel to each other.
If p q and q r, then p r. r q p

47. Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
If m __ p and n __ p, then m p. m n

49. 2.3 Parallel Lines and Triangle Angle Sums

50. Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P

51. Constructing Parallel Lines
Construct a line parallel to line m that passes through point P. WB page 103 P m

52. Perpendicular Postulate
If there is a line and a point not on the line then there is exactly one line through the point perpendicular to the given line.

53. Construction
Construct a line that passes through a given point P and is perpendicular to a given line l.

54. Theorem 4.1 Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180º.
m<A + m<B + m<C = 180º. B
A C

55. Examples:
45º
20º
3x + 16
68º
x º

56. Theorem 4.2 Exterior Angle Theorem
The measure of an exterior angle of a ∆ is equal to the sum of the measures of the two remote interior angles.
Exterior < - formed by one side of a ∆ and the extension of another side. C
A B D

57. Remote interior <‘s - the interior angles of the ∆ not adjacent to a given exterior angle.
A B D

58. Example 1: 56 X

59. Example 2:
65º
82º
46º º 5

60. 5 2 70
118 3 4 36 65 6 1 82 68

61. Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
m<A + m<B = 90º. C
A B

62. FIND X: 50 xº
2(x+4)º

63. Classify the triangle by its angles and by its sides.
Find the measure of the exterior angle shown. xº G
(3x + 10)º
54º
70º
36º
H I

64. Corollary
There can be at most one right or obtuse angle in a triangle.

66. 2.4 Slopes of Parallel and Perpendicular Lines

67. Parallel Lines in the Coordinate Plane
CC.9-12.G.GPE.6 The student will learn to find slopes of lines and use slope to identify parallel lines in a coordinate plane. The student will learn to write equations of parallel lines in a coordinate plane.

68. Postulate 17: Slopes of parallel lines.
In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

69. Slope of a Line
Slope = rise/run = (y - y)/(x - x) 2 1 2 1

70. 1. Find the slope of the line that passes through the points (-4, -3)and (5, 3).
Slope? (-7, 12) and (-7, -4)
Slope? (-6, 8) and (3, 8)

71. Equation of a Line
Slope-Intercept form: y = mx + b Point-slope form: y – y = m(x – x ) 1 1

72. Equation of a Line
Write an equation of the line with the given information.
Slope = 4, point on line (3, -5)
Slope = - ½, point on line (-7, -2)
Slope = 0, point on line (-2, -7)
Slope is undefined, point on line (-4,3)
Points on line: (-5, 9) and (-4, 7)
Points on line: (-20, -10) and (5, 15)

73. Find the slope of each line. Which lines are parallel?
Line j: (-1, 10) & (-5, 2)
Line k: (2, 9) & (-2, 1)
Line l: (5, 7) & (2, 2)

74. Writing equation of a parallel line.
1. Line k has the equation y = -2x + 5.
Line l is parallel to k and passes through the point (-4, 3). Write an equation of line l.

75. Write an equation of the line through (-4, 6) and parallel to the line y = 3x + 8.

76. Write and equation of the line through (-2, 4) and parallel to the line through (1, 1) and (5, 7)

77. Write and equation of the line that passes through (4, 6) and is parallel to the line that passes through (6, -6) and (10, -4)

78. Write an equation of the line through the origin and parallel to the line x – 3y = 9.

79. Perpendicular Lines in the Coordinate Plane
Students apply what they know about slopes of parallel lines and perpendicular lines to determine whether two lines are parallel, perpendicular, or neither. Students will learn to write equation of perpendicular lines.

80. Postulate 18: Slopes of perpendicular lines
In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular.

81. slope perpendicular slope 3 ______ ½ ______ -7 ______ - 4/5 ______ 1 ______ 0 ----------

82. Decide whether j is perpendicular to k.
Line J: (-2, 2) & (2, 0)
Line K: (2, 0) & (5, 6)
Decide whether the lines are perpendicular:
1. 3x – 2y = 1 & 6x + 9y = 3
2. 6x + 2y = 8 & y = -3x - 4

83. Line l has equation y = 3x + 5
Line l has equation y = 3x Find an equation of the line m that passes through P(3, 1) and is perpendicular to l.
Through (2, -2) and perpendicular
to y = (1/4)x + 10.

84. Write an equation of the line that passes through (8, -2) and is perpendicular to the line Y = 7 – 2x.

85. Write an equation of the line that passes through (-3, 5) and is perpendicular to the line that passes through (0, 5) and (6, 1).

86. Write an equation of the line that passes through (6, -10) and is perpendicular to the line that passes through (4, -6) and (3, -4)

87. Write an equation of the line that is the perpendicular bisector of the segment joining (2, 4) and (4, -4)