1. Parallel Lines cut by a Transversal

2. Recall that parallel lines are
two coplanar lines
that never intersect.
AB || CD

3. EXAMPLES: In each case of examples , t is the transversal
A transversal is a line, segment or ray that intersects two or more lines at different distinct points. t
EXAMPLES: In each case of examples , t is the transversal w t v r t n m p q
NON-EXAMPLES: In the non-example case, t is NOT a transversal d c t a
Only one line
Same point>>>
Do not intersect t b t

4. When parallel lines are cut by a transversal,
Several special types of pairs of angles are formed. They are named based on the angles positions relative to the parallel lines and the transversal.

5. When parallel lines are cut by a transversal, eight angles are formed

6. Each parallel line intersects with the transversal to create
two sets of 4 angles.
SET 1
SET 2

7. The top set of angles (1,2,3,and 4) formed have exactly the same measures as the bottom set ( 5,6,7, and 8) when m and n are parallel because they are being cut by the same line (t) at the same angle.

8. The top set of angles (1,2,3,and 4) could be cut and placed on top of the bottom set (5,6,7,and 8)

9. Four sets of corresponding angles are formed.
Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.
1 corresponds to 5
because they are both in the top left position.

10. Four sets of corresponding angles are formed.
Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.
2 corresponds to 6
because they both lie in the top right corner of the sets.

11. Four sets of corresponding angles are formed.
Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.
3 corresponds to 7
because they both lie in the bottom left corner of the sets

12. Four sets of corresponding angles are formed.
Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.
4 corresponds to 8
because they both lie in the bottom right corner of the sets.

13. Corresponding Angles Theorem
If two parallel lines are cut by a transversal, corresponding angles are congruent
m || n
Therefore we know
<1 ≅ <5
<2 ≅ <6
<3 ≅ <7
<4 ≅ <8 3 4 8 7

14. m || n Find the measure of angle 2 Bottom right - Corresponding angles
2x corresponds to 5x + 55
Therefore 2x+100 = 5x + 55
45 = 3x
15 = x
5x +55 =5(15)+55 = 130o
5x +55 and <2 are linear pair so
5x <2 = 180o
<2 = 180o
<2 = 50o

15. m || n When parallel lines are cut by a transversal,
eight angles are formed
The parallel lines, m and n, cut two areas in the plane
called the interior……
m || n

16. m || n When parallel lines are cut by a transversal,
eight angles are formed
The parallel lines, m and n, cut two areas in the plane
called the interior and the exterior
m || n

17. m || n The angles between the parallel lines ( m and n) are
INTERIOR ANGLES
Angles 3 , 4, 5 , and 6 are interior angles.
m || n

18. m || n The angles outside the parallel lines ( m and n ) are
EXTERIOR ANGLES
Angles 1,2, 7, and 8 are exterior angles
m || n

19. The transversal, t , cuts the plane into two regions.
m || n

20. m || n The transversal, t , cuts the plane into two regions.
Angles 2, 4, 6, and 8 lie on the same side of the transversal.
m || n

21. m || n The transversal, t , cuts the plane into two regions.
Angles 1, 3, 5, and 7 are on the same side of the transversal.
m || n

22. m || n The transversal, t , cuts the plane into two regions.
Angles 3 and 6 are on “opposite” sides or
“alternate” sides of the transversal.
m || n

23. m || n Opposite angles fall on alternate sides of the transversal
Other examples of pairs of opposite angles are
3 and 6
8 and 1
7 and 2
5 and 4
m || n

24. Name two pairs of alternate interior angles
m || n
<c and <e
<d and <f

25. Name two pair of same- side interior angles
m || n
<c and <f
<d and <e

26. Name two pairs of alternate exterior angles
m || n
<a and <g
<b and <h

27. Name two pairs of same-side exterior angles
m || n
<a and <h
<b and <g

28. Name four pairs of corresponding angles
m || n
<a and <e
<b and <f
<c and <g
<d and <h

29. Name four pairs of vertical angles
m || n
<a and <c
<b and <d
<e and <g
<f and <h

30. Name eight pairs of supplementary angles
m || n
<a and <b
<b and <c
<c and <d
<d and <a
<e and <f
<f and <g
<g and <h
<h and <e

31. Alternate Interior Angles Theorem

32. GIVEN : p || q
Prove : <1 ≅ <2 p 1 2 3 q t

37. Same-Side Exterior Angles Theorem
a || b
If two parallel lines are cut by a transversal, then same-side exterior angles are supplementary.
< 1 + <3 = 180o

38. Same-Side Exterior Angles Theorem
GIVEN: l || m
PROVE: m<4 +m<5 = 180o 4 5
STATEMENTS
REASONS
l || m
1. Given
<6 ≅ <4 ,
2. Corresponding <‘s Theorem
3. <6 + <5 = 180O
3. Defn. of linear pair
4. <4 + <5 = 180O
4. Substitution Property

39. When 2 || lines are cut by a transversal, Congruent Pairs of angles
Vertical Angles
Corresponding Angles
Alternate Exterior Angles
Alternate Interior Angles

40. Supplementary Pairs of angles
When 2 || lines are cut by a transversal,
Supplementary Pairs of angles
Linear Pair
Same-Side Interior Angles
Same-Side Exterior Angles

41. HOMEWORK
Complete the problems
on the following pages

42. Given: a || b and c || d < 9 = 81o
Find the measures of all the angles. Justify your reasoning using pairs of special angles.
81o

43. Given: a || b and c || d < 9 = 81o
Find the measures of all the angles. Justify your reasoning using pairs of special angles.
81o
99o
81o
99o
99o
81o
99o
81o
81o
99o
81o
99o
99o
99o
81o
81o

44. 17) Solve for x and find the measures of the angles

45. 17) Solve for x and find the measures of the angles
Define the relationship -
same-side interior angles so they are
supplementary ----SUM IS 180o
X+75 + x+125 = 180
2x = 180
2x = -20
x = -10
3.) Substitute the value of x in the
expressions and find the
measures of the angles
to answer the question
2.) Write equation and solve for x
= 65
= 115

46. 18) Solve for x and find the measures of the angles

47. 18) Solve for x and find the measures of the angles
Define the relationship -
corresponding angles so they are
congruent – measures are equal.
12x + 3 = 11x + 9
x = 6
2.) Write equation and solve for x
3.) Substitute the value of x in the
expressions and find the
measures of the angles
to answer the question
12x + 3 = 12(6)+ 3 = 75o
11x + 9 = 11(6) + 9 = 75o

48. 19) Solve for x and find the measures of the angles

49. 19) Solve for x and find the measures of the angles
Define the relationship -
( 15x -5)o and 125o are a linear pair so their
sum is 180o .
(7y +27) o and 125o are alternate exterior
angles so they are congruent.
2.) Write equation and solve for x
( 15x -5)o + 125o= 180o (7y +27) o = 125o
15x +120 = 180o y = 98o
15x = 60o y = 14o
x = 4
15x -25 = 15(4) -25 = y +27 = 125o
7(14) +27 = 125o
125o = 125o
3.) Substitute the value of x in the
expressions and find the
measures of the angles
to answer the question

50. Solve for x and find the measures of the angles
AB || CD

51. Solve for x and find the measures of the angles AB || CD
1.) Alternate interior angles are congruent
2.) = 3x
40 = x
3.) (x) = 3(40) = 120

52. Solve for x and y.
Then find the measures of the angles

53. Then find the measures of the angles
Solve for x and y.
Then find the measures of the angles
1.) (6x +y) and ( x + 5y) are corresponding angles
so they are congruent.
4x and ( 6x + y ) are a linear pair so they are
supplementary and have a sum of 180.
2.) x +y = x + 5y and x + 6x + Y = 180
5X = 4Y X +y = 180
y = 180 – 10x
Since there are 2 variables in both equations, you have a system Y = 180 – 10x
5x = 4y
Solve by substitution x = 4 ( 180 – 10x) (16) = 4y
5x = 720 – 40x = 4y
45x = = y
x = 16
3.) 4x = 4(16) = 64O x + Y = 6 (16) + 20= 116 O X+5Y = 16 +5(20) = 116O

54. 22.) Solve for x and y.
Then find the measures of the angles

55. Then find the measures of the angles
22.) Solve for x and y.
Then find the measures of the angles
1.) Same Side Interior Angles Supplementary
3y y+15 = 180
Linear Pair =180
5y x+5 = 180
2.) Simplify and solve to find the variables
3y y+15 = y x+5 = 180
8y = y +2X +20 = 180
8Y = Y + 2X = 160
Y = (20) +2X = 160
X = 160
2X = 60
X = 30
VERTICAL ANGLES
17X -70 = 17(30)-70 = 440
3Y +5 = 3(20) +5 = = 65
5Y+15 = 5(20) +15 = 115
2X+5 = 2(30)

56. 23) Solve for x and find the measures of the angles

57. 23) Solve for x and find the measures of the angles
Alternate exterior angles are =
X2 -2X -5 = 2X2 – 7X -19
0 = X2 – 5x -14
0 = ( x -7)(x +2) so……
x = 7 or x = -2
If x = 7…………… If x = -2……………
X2 -2X = (7)2 -2(7) = X2 -2X -5 = (-2)2 -2(-2) -5= 3
2X2 – 7X -19 =2(7)2 –7(7) -19 = X2 – 7X -19= 2(-2)2 –7(-2) -19 = 3

58. Solve for x and y.
Then find the measures of the angles.
c||d and a|| b

59. Then find the measures of the angles. c||d and a|| b
Solve for x and y.
Then find the measures of the angles.
c||d and a|| b 87 93 87
Alternate
Interior Angles = 87
Linear Pair have
sum of 180
2x+13 = 3x -24
37 = x
2(37)+13 = 87
3(37) -24 = 87
3y = 180
3y = 180
3y = 69
y = 23
3y + 24 = 3(23) + 24 = 93
4y – 5 = 4( 23) – 5 = 87

60. 25.) Solve for all the variables.
Then find the measures of the angles.

61. 25.) Solve for all the variables.
Then find the measures of the angles.
ONE VARIABLE!
Same side interior angles are congruent
A + 30 = 60
A = 30 60

62. Corresponding angles =60
30 +2b = 60
2b = 30
b = 13

63. Vertical angles 25.) Solve for all the variables.
Then find the measures of the angles.
Vertical angles 60 60
5b – 5c = 60
5(13) -5c = 60
c = 60
-5c = -5
C = 1 60
b=13

64. Alternate Interior Angles =
25.) Solve for all the variables.
Then find the measures of the angles.
C=1
10c + d = 60
10(1) +d = 60
10 + d = 60
d = 50 60 60 60 60
Alternate Interior Angles =

65. 25.) Solve for all the variables.
Then find the measures of the angles.
Linear pair
Sum = 180 60 60
120
12d +6e +60 = 180
12d + 6e = 120
12(1) +6e=120
12 +6e = 120
6e = 108
e = 18 60 60
d = 1

66. 25.) Solve for all the variables.
Then find the measures of the angles.
Alternate
Interior Angles= 60 60
120
4f +4e = 120
4f +4(18) = 120
4f+72 = 120
4f = 48
f = 12 60
120 60
e = 18