GOAL 1 PROPERTIES OF PARALLEL LINES

This section will require you to think about and use parallel lines. Although some of the theorems and ideas may seem obvious, you must always be ready to provide justification for any statements made.

3.3PARALLEL LINES AND TRANSVERSALS

POSTULATE

Corresponding Angles Postulate lines corr. s

THEOREMS

Alternate Interior Angles lines alt. int. s

Consecutive Interior Angles lines cons. int. s supp.

Alternate Exterior Angles lines alt. ext. s

Perpendicular Transversal Transversal one of 2 lines other line

EXAMPLE 1

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

Extra Example 1

p q Given

1 4 Alt. Ext. s Thm.

2 and 4 are supp. Linear Pair Post.

2 4 180m m

1 4m m Def. s

1 2 180m m

Def. supp. s

Substitution

Given:

Prove:

p q

1 2 180m m 4

1

2

35 6

7 8

p

q

EXAMPLE 2

Extra Example 2

4

1

2

35 6

7 8

Given thatfind each measure. Give the postulate or theorem used.

2 110 ,m

8ma.

b.

c.

d.

1m

3m

4m

70°; Linear Pair Postulate

70°; Corresponding Angles Postulate with 8

110°; Alternate Exterior Angles Thm. or

with 2Linear Pair Postulate with 1

70°; Alternate Exterior Angles Thm. or

with 1 Linear Pair Postulateor

with 2Vertical Angles Theorem with 8

Checkpoint

EXAMPLE 3

A B

DC

1

2

3

100°Find 1, 2, and 3.m m m

1 80 ; 2 100 , 3 80m m m

Extra Example 3In the diagram above, how many angles have a measure of 100°?

eight

GOAL 2 PROPERTIES OF SPECIAL PAIRS OF ANGLES

EXAMPLE 4

3.3PARALLEL LINES AND TRANSVERSALS

Extra Example 4

EXAMPLE 5

Use properties of parallel linesto find the value of x.

72° 1(x – 8 )°

Since by the Vertical Angles Theorem, and by the Consecutive Interior Angles

Theorem, solve the equation:

1 72m 1 ( 8) 180m x

72 ( 8) 180x

116x

Extra Example 5Refer to Example 5. We now know that the diameter of Earth is about 7973 mi. Recalculate the distance between Syene and Alexandria using this figure.

Using the equation from Example 5:

1 of a circle =

50 Earth's circumference

d

1(7973 miles)

50d

501 miles d

CheckpointUse properties of parallel lines to find the value of x.

(x – 20)°x°

70° m

n

x = 65

QUESTION:

If a transversal is perpendicular to one of two parallel lines, what is the measure of all the

angles formed?

ANSWER:

90°