1. Parallel Lines and Transversals
Lesson 3.4 w r 8 2 4 3 5 6 7 1
r || w

2. Objective:
Find the congruent angles formed when a transversal cuts parallel lines.

3. Key Vocabulary
None

4. Postulates
8 - Corresponding Angles

5. Theorems 3.5 Alternate Interior Angles 3.6 Alternate Exterior Angles
3.7 Same-Side Interior Angles

6. Parallel Lines and Angle Pairs
Line 𝓏 is a transversal of parallel lines 𝓍 and 𝓎.
Since lines 𝓍 and 𝓎 are parallel, there are special relationships between specific pairs of angles.

7. Parallel Lines Transversal
Review:
Parallel Lines
Transversal

8. PARALLEL LINES
Two or more lines are parallel if and only if they are in the same plane and they do not intersect. (line w and r) w r
r || w

9. TRANSVERSAL
A line intersecting two or more coplanar lines. (lines r and w) r w
r || w

10. Postulate 8 Corresponding ’s Postulate
If 2 parallel lines are cut by a transversal, then each pair of corresponding ’s is .
i.e. If l m, then 12. 1 2 l m

11. Corresponding Angles
Look for angles in an F shape to help you find corresponding angles.

12. CORRESPONDING ANGLES
If two parallel lines are cut by a transversal, then Corresponding angles are congruent. r w
r || w 1 5 2 6 3 7 4 8

13. Example 1 Find the measure of the numbered angle. SOLUTION a.c.
SOLUTION a.
m6 = 60° b.
m5 = 135° c.
m2 = 90° 13

14. Your Turn: Find the measure of the numbered angle. ANSWER 120° ANSWER1.
ANSWER
120° 2.
ANSWER
145° 3.
ANSWER
45°

15. Example 2: In the figure and Find Corresponding Angles Postulate
Vertical Angles Postulate
Transitive Property
Definition of congruent angles
Substitution
Answer:

16. Your Turn:
In the figure and Find
Answer:

17. Theorem 3.5 Alternate Interior ’s Theorem
If 2 parallel lines are cut by a transversal, then each pair of alternate interior ’s is .
i.e. If l m, then 12. 1 2 l m

18. Alternate Interior Angles
Look for angles inside a Z or N shape to find alternate interior angles.

19. ALTERNATE INTERIOR ANGELS
If two parallel lines are cut by a transversal, then Alternate Interior angles are congruent. w r
r || w 3 4 5 6

20. Example 3 Find the measure of PQR. SOLUTION a. mPQR = 35° b.c.
SOLUTION a.
mPQR = 35° b.
mPQR = 120° c.
mPQR = 70° 20

21. Your Turn: Find the measure of the numbered angle. ANSWER 90° ANSWER1.
ANSWER
90° 2.
ANSWER
65° 3.
ANSWER
100°

22. Theorem 3.6 Alternate Exterior ’s Theorem
If 2 parallel lines are cut by a transversal, then the pairs of alternate exterior ’s are .
i.e. If l m, then 12.
l m 1 2

23. ALTERNATE EXTERIOR ANGLES
If two parallel lines are cut by a transversal, then Alternate Exterior Angles are congruent. w r
r || w 1 2 7 8

24. Example 4 Linear Pair Postulate Substitute 75° for m2.
Find the measures of 1 and 2.
SOLUTION
The measure of 2 is 75° because alternate exterior angles are congruent. The measure of 2 can be used to find the measure of 1.
m1 + m2 = 180°
Linear Pair Postulate
m1 + 75° = 180°
Substitute 75° for m2.
m1 + 75° – 75° = 180° – 75°
Subtract 75° from each side.
m1 = 105°
Simplify. 24

25. Your Turn: Find the measure of the numbered angle. ANSWER 130° ANSWER1.
ANSWER
130° 2.
ANSWER
42° 3.
ANSWER
90°

26. Your Turn:
Use the diagram. Tell whether the angles are congruent or not congruent. Explain.
ANSWER
congruent by the Alternate Exterior Angles Theorem 4.
1 and 8
ANSWER
Not congruent; the angles are a linear pair. 5.
3 and 4
ANSWER
Not congruent; the angles are a linear pair. 6.
4 and 2

27. Your Turn:
Use the diagram. Tell whether the angles are congruent or not congruent. Explain.
ANSWER
congruent by the Alternate Exterior Angles Theorem 7.
2 and 7
ANSWER
congruent by the Corresponding Angles Postulate 8.
3 and 7
ANSWER
Not congruent; there is no special relationship between these angles. 9.
3 and 8

28. Theorem 3.7 Same-Side Interior ’s Theorem
If 2 parallel lines are cut by a transversal, then each pair of same-side interior ’s is supplementary.
i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°. l m 1 2

29. Same-Side Interior Angles
Look for angles inside a C shape to find same-side interior angles.

30. Same-side INTERIOR ANGELS
If two parallel lines are cut by a transversal, then each pair of Same-Side Interior Angles is supplementary. w r
r || w 3 4 5 6

31. Example 5 Find the measure of the numbered angle. SOLUTION a.31

32. Example 6 Find the value of x. Corresponding Angles Postulate
SOLUTION
(x + 15)° = 125°
Corresponding Angles Postulate
x = 110
Subtract 15 from each side. 32

33. Your Turn: Find the value of x. 1. ANSWER 85 2. ANSWER 104 3. ANSWER40

34. How to Find Angle Measurements
On Two Parallel Lines Cut By A Transversal w r 8 2 4 3 5 6 7 1

35. PARALLEL LINES If two parallel lines are cut by a transversal, then ……
Corresponding angles are congruent,
Alternate Interior angles are congruent, And
Alternate Exterior angles are congruent. lw lr 4 1 3 2 5 6 7 8 1 5 1 4 5 8

36. Corresponding
and
Vertical
Angles r w
r || w 2 3 4 6 5 7 8
58 ˚
58 ˚
58 ˚
58˚
58 ˚
58˚
If angle 1 = 58 ˚ then angle 5 = 58 ˚ because they are corresponding
angles, which are congruent to each other
Since angle 1 = 58 ˚ then angle 4 = 58 ˚ and since angle 5 = 58 ˚ then
angle 8 = 58 ˚ because they are vertical angles, which are congruent to
each other

37. Alternate
Exterior
and
Interior
Angles r w
r || w 2 3 4 6 5 7 8
58 ˚
122˚
122 ˚
122 ˚
122˚
If angle 1 = 58 ˚ then angle 2 = 122 ˚ because the two angles form a line, which is equal to 180 ˚
Since angle 2 = 122 ˚ then angle 7 = 122 ˚ because they are Alternate Exterior angles, which are congruent to each other.
Since angle 3 = 122 ˚ then angle 6 = 122 ˚ because they are Alternate Interior angles, which are congruent to each other.

38. Example 7:
What is the measure of RTV?

39. Example 7: Alternate Interior Angles Theorem
Definition of congruent angles
Substitution

40. Example 7: Alternate Interior Angles Theorem
Definition of congruent angles
Substitution
Angle Addition Postulate
Answer: RTV = 125°

41. Example 8: If ALGEBRA and find x and y. Find x.
by the Corresponding Angles Postulate.

42. Example 8: Definition of congruent angles Substitution
Subtract x from each side and add 10 to each side.
Find y.
by the Alternate Exterior Angles Theorem.
Definition of congruent angles
Substitution

43. Example 8: Simplify. Add 100 to each side. Divide each side by 4.
Answer:

44. Your Turn:
ALGEBRA If
and
find x and y.
Answer:

45. Example 9: 55° 125° 40° m1 = m2 = m3 = m4 = m5 = m6 = x = Find:
125o 2
Find:
m1 =
m2 =
m3 =
m4 =
m5 =
m6 =
x =
55°
125°
40° 3 5
x+15o

46. Joke Time What flower grows between your nose and your chin? Tulips
How many sides are there to a circle?
2 – inside and outside.
What do you get when you cross an elephant and Darth Vader?
An elevader.

47. Assignment
Section 3.4, pg : #1-12 all, odd