Parallel Lines and Transversals

2. A Line That Intersects 2 Or MoreA Line That Intersects 2 Or More
Lines At Different Points IsLines At Different Points Is
Called A TransversalCalled A Transversal
transversaltransversal

3. When This Happens,When This Happens,
8 Angles Are Formed8 Angles Are Formed
11
22
33
44
55
66
77
88

4. 11
22
33
44
55
66
77
88
This Forms 2 NeighborhoodsThis Forms 2 Neighborhoods

5. 11
22
33
44
55
66
77
88
Remember:Remember:
Vertical And Linear Pair AnglesVertical And Linear Pair Angles
VerticalVertical

6. Linear PairsLinear Pairs
11
22
33
44
55
66
77
88

7. 11 33 55 77
22 44 66 88
These Angles AreThese Angles Are
Called ConsecutiveCalled Consecutive
Or Same Side AnglesOr Same Side Angles

8. 33
44
55
66
11
22
77
88
Interior AnglesInterior Angles
(Between 2 lines)(Between 2 lines)
Exterior AnglesExterior Angles
((outside the lines)outside the lines)

9. Alternate Angles Are OnAlternate Angles Are On
Different Sides Of The TransversalDifferent Sides Of The Transversal
AndAnd From Different NeighborhoodsFrom Different Neighborhoods
11
22
33
44
55
66
77
88
Alternate ExteriorAlternate Exterior
Angles 1 And 8Angles 1 And 8
Angles 2 And 7Angles 2 And 7
Alternate InteriorAlternate Interior
Angles 3 And 6Angles 3 And 6
Angles 4 And 5Angles 4 And 5

10. 11
33 55
77
22
44 66
88
Consecutive IntConsecutive Int
Angles 3 and 5Angles 3 and 5
Angles 4 and 6Angles 4 and 6
Consecutive ExtConsecutive Ext
Angles 1 and 7Angles 1 and 7
Angles 2 and 8Angles 2 and 8

11. 11
22
33
44
55
66
77
88
Corresponding Angles AreCorresponding Angles Are
Located In The Same Position InLocated In The Same Position In
Each NeighborhoodEach Neighborhood

12. ANSWER alternate interior
1. 5, 7
2. 3, 6
3. 1, 8
ANSWER alternate exterior
ANSWER corresponding
Identify the type of angles.

13. 1111 1212
1313 1414
1515 1616
1717 1818
Name The AnglesName The Angles
1.1. 11 and 1511 and 15
2.2. 12 and 1612 and 16
3.3. 13 and 1613 and 16
4.4. 12 and 1812 and 18
5.5. 14 and 1614 and 16
6.6. 14 and 1814 and 18
7.7. 11 and 1411 and 14
8.8. 15 and 1715 and 17

14. 1.1.CorrespondingCorresponding
2.2.CorrespondingCorresponding
3.3.Alt InteriorAlt Interior
4.4.Consecutive (SS) ExteriorConsecutive (SS) Exterior
5.5.Consecutive (SS) InteriorConsecutive (SS) Interior
6.6.CorrespondingCorresponding
7.7.VerticalVertical
8.8.Linear PairLinear Pair

15. 11 22
33 44
55 66 77 88
99 1010
1111 1212
1313 1414
1515 1616
With This Diagram, We Can Work WithWith This Diagram, We Can Work With
Angles In Different Neighborhoods As LongAngles In Different Neighborhoods As Long
As They Are Connected By A TransversalAs They Are Connected By A Transversal
Name the anglesName the angles
1.1. 1 and 31 and 3
2.2. 7 and 127 and 12
3.3. 11 and 1411 and 14
4.4. 6 and 106 and 10
5.5. 13 and 513 and 5
6.6. 9 and 69 and 6
7.7. 1 and 131 and 13
8.8. 5 and 45 and 4
9.9. 7 and 117 and 11
10.10. 6 and 116 and 11

16. 1.1. CorrespondingCorresponding
2.2. Alt. Int.Alt. Int.
3.3. Alt. Int.Alt. Int.
4.4. Cons. (SS) Int.Cons. (SS) Int.
5.5. CorrespondingCorresponding
6.6. Alt. Int.Alt. Int.
7.7. Consecutive ExtConsecutive Ext
8.8. Alt. ExtAlt. Ext
9.9. Cons. (SS) Int.Cons. (SS) Int.
10.10.NoneNone

17. 2. 2 and 10 are
alternate interior
angles.
Parallel Lines and Transversals
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
Determine if the statement is true or false.
If false, correct the statement.
1. Line r is a
transversal of lines p
and q.
∠

18. 4. < 1 and < 15 are
alternate exterior
angles.
3. <3 and < 5 are
alternate interior
angles.
Parallel Lines and Transversals
∠
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
Determine if the statement is true or false.
If false, correct the statement.
∠
∠

19. 6. < 10 and < 11 are
consecutive interior
angles.
5. < 6 and < 12 are
alternate interior
angles.
Parallel Lines and Transversals
∠
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
Determine if the statement is true or false. If
false, correct the statement.
∠
∠

20. Determine if the statement is true or false.
If false, correct the statement.
7. < 3 and < 4 are
alternate exterior
angles.
8. < 16 and < 14 are
corresponding angles.
Parallel Lines and Transversals
∠
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
∠
∠

21. 2. < 2 and < 10 are
alternate interior
angles.
Parallel Lines and Transversals
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
Determine if the statement is true or false.
If false, correct the statement.
1. Line r is a
transversal of lines p
and q.
True – Line r
intersects both lines
in a plane.
False - The angles are
corresponding angles on
transversal p.

22. 4. < 1 and < 15 are
alternate exterior
angles.
3. < 3 and < 5 are alternate
interior angles.
Parallel Lines and Transversals
∠
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
Determine if the statement is true or false.
If false, correct the statement.
False – The angles are
vertical angles created
by the intersection of
q and r.
∠
True - The angles are
alternate exterior
angles on transversal p.
∠

23. 6. < 10 and < 11 are
consecutive interior
angles.
5. < 6 and < 12 are
alternate interior
angles.
Parallel Lines and Transversals
∠
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
Determine if the statement is true or false.
If false, correct the statement.
True – The angles are
alternate interior angles
on transversal q.
∠
True – The angles are
consecutive interior
angles on transversal
s.
∠

24. Determine if the statement is true or false.
If false, correct the statement.
7. < 3 and < 4 are
alternate exterior
angles.
8. < 16 and < 14 are
corresponding angles.
Parallel Lines and Transversals
∠
∠
Identifying Angles – Check for Understanding
1 2 3 4
5678
9 10 11 12
13141516
∠
∠
False– The angles are
adjacent angles on
transversal r.
True – The angles
are corresponding
angles on transversal
s.

26. If 2 Parallel Lines Are Cut By AIf 2 Parallel Lines Are Cut By A
Transversal Then:Transversal Then:
Corresponding AnglesCorresponding Angles
Are CongruentAre Congruent
Alternate InteriorAlternate Interior
Angles Are CongruentAngles Are Congruent
Same Side Interior AnglesSame Side Interior Angles
Are SupplementaryAre Supplementary

27. Remember ………
Even Without Parallel Lines
Vertical Angles Are Always
Congruent
Linear Pairs Are Always
Supplementary

28. Example 1
SOLUTION
By the Corresponding Angles Postulate, m 5 =
120°. Using the Vertical Angles Congruence
Theorem, m 4 = 120°. Because 4 and 8
are corresponding angles, by the Corresponding
Angles Postulate, you know that m 8 = 120°.
The measure of three of
the numbered angles is
120°. Identify the angles.
Explain your reasoning.

29. Example 2
Use the diagram.
1. If m 1 = 105°, find
m 4, m 5, and m 8.
Tell which postulate or
theorem you use in each
case.
Vertical Angles Congruence Theorem.
Corresponding Angles Postulate.m 5 =105°
Alternate Exterior Angles Theoremm 8 =105°
m 4 =105°
ANSWER

30. 11 22
33 44
55 66
77 88
aa
bb
a ba b
m 1 = 105m 1 = 105
Find:Find:
1.1. 3 =3 =
2.2. 6 =6 =
3.3. 7 =7 =
4.4. 4 =4 =
5.5. 5 =5 =
7575
7575
7575
105105
105105

31. 11 22
33 44
55 66
77 88
aa
bb
63°63°
117°117°
119°119°
119°119°
119°119°
119°119°
61°61°
63°63°
63°63°

32. Use the diagram.
If m 3 = 68° and
m 8 = (2x + 4)°, what is
the value of x? Show
your steps.
2.
Guided Practice
m 3 = m 7
68 + 2x + 4 = 180
2x + 72 = 180
2x = 108
x = 54
m 7 + m 8 =180
ANSWER

33. Example 3
ALGEBRA
Find the value of x.
SOLUTION
By the Vertical Angles Congruence Theorem,
m 4 = 115°. Lines a and b are parallel, so you
can use the theorems about parallel lines.
Consecutive Interior Angles Theoremm 4 + (x+5)° = 180°
Substitute 115° for m 4.115° + (x+5)°= 180°
Combine like terms.x + 120 = 180
Subtract 120 from each side.x = 60

34. a ba b 2x+62x+6
3x-103x-10
5x-205x-20
2x-102x-10
2x+6 = 3x-102x+6 = 3x-10
6 = x – 106 = x – 10
16 = x16 = x
5x-20+2x-10 = 1805x-20+2x-10 = 180
7x-30 = 1807x-30 = 180
7x = 2107x = 210
x = 30x = 30
4x+254x+25
6x-156x-15
4x+25 = 6x-154x+25 = 6x-15
25 = 2x-1525 = 2x-15
40 = 2x40 = 2x
20 = x20 = x