# Parallel And Perpendicular Lines Worksheet Page 29

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