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The construction establishes that there is at least one line through P that is parallel

to MN. In 1795, Scottish physicist and mathematician John Playfair provided the

modern version of Euclid’s Parallel Postulate, which states there is exactly one line

parallel to a line through a given point not on the line.

Postulate 3.5

Postulate 3.5

Parallel Postulate

If given a line and a point not on the line, then there exists

exactly one line through the point that is parallel to the given line.

Parallel lines with a transversal create many pairs of congruent angles. Conversely,

those pairs of congruent angles can determine whether a pair of lines is parallel.

Proving Lines Parallel

Theorems

Examples

3.5

If two lines in a plane are cut by a transversal so that

If

1

8 or if

m n

a pair of alternate exterior angles is congruent, then

2

7, then

.

the two lines are parallel.

Abbreviation:

If alt. ext.

are

, then lines are .

m

1

2

3.6

3

4

If two lines in a plane are cut by a transversal so that

If m 3

m 5

180

a pair of consecutive interior angles is supplementary,

or if m 4

m 6

n

5

6

m n

then the lines are parallel.

180, then

.

7

8

If cons. int.

are suppl., then lines are .

Abbreviation:

3.7

If two lines in a plane are cut by a transversal so that a

If

3

6 or if

m n

pair of alternate interior angles is congruent, then the

4

5, then

.

lines are parallel.

If alt. int.

are

, then lines are .

Abbreviation:

3.8

m

n

In a plane, if two lines are perpendicular to the same line,

If

and

,

m

m n

then they are parallel.

then

.

n

Abbreviation:

If 2 lines are

to the same line, then lines are .

Identify Parallel Lines

Example

Example

1

1

In the figure, BG bisects

ABH. Determine

which lines, if any, are parallel.

A

• The sum of the angle measures

B

in a triangle must be 180, so

˚

45

m BDF

180

(45

65) or 70.

˚

D

65

F

• Since

BDF and

BGH have the same

measure, they are congruent.

• Congruent corresponding angles

˚

70

G

H

indicate parallel lines. So, DF GH.

•

ABD

DBF, because BG bisects

ABH. So, m ABD

45.

•

ABD and

BDF are alternate interior

angles, but they have different

measures so they are not congruent.

• Thus, AB is not parallel to DF or GH.

152 Chapter 3 Parallel and Perpendicular Lines

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