Chapter 3
Parallel and Perpendicular Lines
3.1 Identify Pairs of Lines and Angles

Parallel lines- ( II ) do not intersect and are coplanar


Parallel planes- do not intersect
Skew lines- do not intersect and are NOT coplanar
Parallel Postulate

If there is a line and a point not on the line, then there is
exactly one line through the point parallel to the given
line.
P
k
Perpendicular Postulate

If there is a line and a point not on the line, then there is
exactly one line through the point perpendicular to the
given line.
P
k

Transversal- a line that intersects two or more
coplanar lines at different points
t
m
k




Corresponding angles- 2 angles in the same
corresponding position relative to the 2 lines
Alternate interior angles- 2 nonadjacent interior
angles on opposite sides of the transversal
Alternate exterior angles- 2 exterior angles on
opposite sides of the transversal
Consecutive interior angles (same side interior)- 2
interior angles on the same side of the transversal

Corresponding

Alternate interior

Alternate exterior

Consecutive interior
t
2
1
3
5 6
7 8
m
4
k
Example
p
1
3
9 10
11 12
5 6
7 8
m
2
4
k
3.2 Using Parallel Lines and
Transversals

Corresponding Angles Postulate- If 2 parallel lines
are cut by a transversal, then corresponding angles are
congruent.
t
1 2
3 4
5 6
7 8
j
k


Alternate Interior Angles TheoremIf 2 parallel lines are cut by a transversal,
then alternate interior
t
angles are congruent.
1 2
Alternate Exterior Angles
3 4
Theorem- If 2 parallel
lines are cut by a transversal,
5 6
then alternate exterior angles
7 8
are congruent.
j
k

Consecutive Interior Angles Theorem- If 2 parallel
lines are cut by a transversal, then consecutive interior
angles are supplementary.
t
1 2
3 4
5 6
7 8
j
k

Perpendicular Transversal Theorem- If a transversal
is perpendicular to one of two parallel lines, then it is
perpendicular to the other one also.
t
1
2
j
k
Prove the Alternate Interior Angles
Theorem
t
Given: j ll k
Prove: 12
3
1
2
j
k
Example
Find m1
70
1
125
3.3 Proving Lines are Parallel

Corresponding Angles Converse (Postulate)If
two lines are cut by a transversal so the corresponding
angles are congruent, then the lines are parallel.
Theorems


Alternate Interior Angles Converse - If two lines are
cut by a transversal so the alternate interior angles are
congruent, then the lines are parallel.
Alternate Exterior Angles Converse - If two lines
are cut by a transversal so the alternate exterior angles
are congruent, then the lines are parallel.

Consecutive Interior Angles Converse - If two lines
are cut by a transversal so the consecutive interior angles
are supplementary, then the lines are parallel.

Transitive Property of Parallel Lines- If two lines
are parallel to the same line, then they are parallel to each
other.
2-column proof of
Alternate Interior Angles Converse
t
Given: 4  5
Prove: g ll h
1
g
4
5
h
3.4 Find and Use Slopes of Lines

Slope, m = riseor
run
y2 - y1
6
4
x2 - x1
run
2
rise
-10
-5
x1,y1
x2,y2
5
10
-2
-4
-6

Steeper slopes have higher absolute values.
m=0
+
-
m is
undefined
Example

Line k passes through (0,3) and (5,2). Graph the line
perpendicular to k and passes through the point (1,2).
Find the equation of the perpendicular line.
Example

Graph using the x and y intercept: 3x + 4y = 12
Example



Your gym membership has a one time initial fee and a monthly
fee.
2 months total spent = $ 231
5months
total spent = $ 363

Write an equation in slope intercept
form to represent the cost of the
membership.

What does the slope represent?

What does the y intercept represent?
3.5 Write and Graph Equations of Lines

Write the equation of the line in slope intercept form.
6
4
2
-10
-5
5
-2
-4
-6
10
Example

Write the equation of the line passing through ( -1, 1) and
parallel to y = 2x – 3.
Example

Write the equation of the line passing through ( 2, 3) and
perpendicular to y = ½ x + 2.
Example

Write the equation of the line that is the perpendicular
bisector of the line segment with endpoints:
P( -4, 3 )
Q (4, -1).
3.6 Prove Theorems about Perpendicular
Lines



If 2 lines are perpendicular, then they intersect to form 4
right angles.
If 2 lines intersect to form a linear pair of congruent
angles, then the lines are perpendicular.
If two sides of two adjacent acute angles are
perpendicular, then the angles are complementary.
Find the distance between a point and a line

Line m contains (0, -3) and (7, 4). P (4, 3)
Find the distance between 2 parallel
lines


Line j: y = 5x – 22
Line m: y = 5x + 4