1 2

3 4

5 6

78

Angles and Parallel Lines

2

Transversal

Definition: A line that intersects two or more lines in a plane at different points is called a transversal.

When a transversal t intersects line n and m, EIGHT angles are formed

Exterior angles: Outside the lines

Interior angles : Between the lines

tm

n

3

Vertical Angles & Linear Pair

Vertical Angles:

Linear Pair:

1 4, 2 3, 5 8, 6 7

Two angles that are opposite angles.

Vertical angles are congruent.

1 & 2 , 2 & 4 , 4 &3, 3 & 1, 5 & 6, 6 & 8, 8 & 7, 7 & 5

Supplementary angles that form a straight line (sum = 180)

1 2

3 4

5 67 8

4

Corresponding Angles

Corresponding Angles: Two angles, on the same side of the transversal, that occupy corresponding positions, one interior and one exterior.

2 and 6, 1 and 5, 3 and 7, 4 and 8

1 2

3 4

5 6

7 8

5

Alternate Angles

Alternate Interior Angles: Two angles that lie between the lines on opposite sides of the transversal (but not a linear pair).

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

3 and 6, 4 and 5

2 and 7, 1 and 8

1 2

3 4

5 6

7 8

6

Consecutive Angles

Consecutive Interior Angles: Two angles that lie between the lines, both on the same side of the transversal.

Consecutive Exterior Angles: Two angles that lie outside the lines, both on the same side of the transversal.

3 and 5 , 4 and 6

1 and 7 , 2 and 8 1 2

3 4

5 6

7 8

ExampleList all pairs that fit the description

a. Corresponding

b. Alternate Exterior

c. Alternate Interior

d. Consecutive Interior

< 4 and < 2 < 3 and < 1 < 5 and < 7 < 6 and < 8

< 2 and < 6 < 3 and < 7

< 4 and < 8 < 1 and < 5

< 3 and < 2 < 6 and < 7

ExampleComplete the statement with corresponding, alternate exterior, alternate interior, or consecutive interior.

1. < 4 and < 8 are

2. < 2 and < 6 are

3. < 1 and < 8 are

4. < 7 and < 2 are

5. < 4 and < 6 are

6. < 5 and < 7 are

Alternate interior

Consecutive exterior

Corresponding

Vertical

Alternate exterior

Consecutive interior

Investigation

Let’s check out therelationship of these angle pairs!

10

Angles and Parallel LinesIf two parallel lines are cut by a transversal, then the following

pairs of angles are CONGRUENT.

• Corresponding angles• Alternate interior angles• Alternate exterior angles

If two parallel lines are cut by a transversal, then the following pairs of angles are SUPPLEMENTARY.

• Consecutive interior angles• Consecutive exterior angles

Example

Given a ll b, find each measure given that m < 6 = 67°.

m 6

m 7

m 8

m 9

m 10

m 11

m 12

m 13

Example

State the postulate or theorem that justifies the statement.

3 7

3 6

2 7

m m 4 6 180

13

Example: If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m< 1 = 100°. Justify your answers.

m<2=80° m<3=100° m<4=80°

m<5=100° m<6=80° m<7=100° m<8=80°

m<9=100° m<10=80° m<11=100° m<12=80°

m<13=100° m<14=80° m<15=100° m<16=80°

t

16 151413

12 11109

8 7

65

34

21

s

DC

BA

14

Example:

1. the value of x, if m<3 = 4x + 6 and the m<11 = 126.

If line AB is parallel to line CD and s is parallel to t, find:

2. the value of x, if m<1 = 100 and m<8 = 2x + 10.

3. the value of y, if m<11 = 3y – 5 and m<16 = 2y + 20.

ANSWERS:

t

16 151413

12 11109

8 7

65

34

21

s

DC

BA

1. 30

2. 35

3. 33