The document outlines the concepts of parallel and perpendicular lines, emphasizing the relationship between their slopes. It explains that two lines are parallel if they have the same slope and perpendicular if the product of their slopes equals -1. Additionally, it provides examples and practice problems to determine line relationships and write equations for lines that are parallel or perpendicular to given lines.

1. Parallel & Perpendicular Lines
The student is able to (I can):
• Use slopes to identify parallel and perpendicular lines.
• Write equations of line parallel or perpendicular to a given
line through a given point.

2. Parallel Lines
Theorem
In a coordinate plane, two nonvertical lines
are parallel if and only if they have the
same slope.
Any two vertical lines are parallel
x
y
ms = mt ⇒ s t
s t
s
1 3 4
m 2
2 0 2
− − −
= = =
− − −
t
3 1 4
m 2
1 1 2
− − −
= = =
− − −

3. Perpendicular
Lines Theorem
In a coordinate plane, two nonvertical lines
are perpendicular if and only if the product
of their slopes is —1 (negative reciprocals).
Vertical and horizontal lines are
perpendicular.
x
y
p
q
p
2 4 6
m 3
2 0 2
+
= = = −
− − −
q
0 1 1 1
m
3 0 3 3
− −
= = =
− − −
p qm m 1= − ⇒ ⊥i p q

4. Practice
Given A(—3, —1), B(3, 3), C(—4, 4), and D(0, —2), is AB
parallel or perpendicular to CD?

5. Practice
Given A(—3, —1), B(3, 3), C(—4, 4), and D(0, —2), is AB
parallel or perpendicular to CD?
The two slopes are not equal, so the lines are not
parallel. The product of the slopes is —1, so the lines are
perpendicularperpendicularperpendicularperpendicular.
3 ( 1) 4 2
AB : m
3 ( 3) 6 3
− −
= = =
− −
2 4 6 3
CD : m
0 ( 4) 4 2
− − −
= = = −
− −

6. Pairs of Lines
Two lines will do one of three things:
• Not intersect (parallel)
• Intersect at one point
• Intersect at all points (coincide)

7. • To determine which of these possibilities is true, look at
the slope and y-intercept:
• To compare slopes and y-intercepts, put both equations
in slope-intercept form (y=mx+b). If we do that to the
last equation, we can see why the two coincide:
y — 5 = 3(x — 1)
y = 3x — 3 + 5
y = 3x + 2
Parallel LinesParallel LinesParallel LinesParallel Lines Intersecting LinesIntersecting LinesIntersecting LinesIntersecting Lines Coinciding LinesCoinciding LinesCoinciding LinesCoinciding Lines
y = 2x — 9
y = 2x + 7
y = 3x + 5
y = —4x — 1
y = 3x + 2
y — 5 = 3(x — 1)
same slope,
different
intercept
different slopes
same slope, same
intercept

8. To write the equation of a line that is
parallel (or perpendicular) to a given line
through a given point:
• Determine the slope of the given line
• Determine the slope of the new line
— Parallel lines have the same slope
— Perpendicular lines have slopes that
are the negative reciprocal
• Write the new equation in point-slope
form
• Solve for y if necessary

9. Example: Write the equation of the line
that is parallel to x − 3y = 15 through the
point (−3, 2).

10. Example: Write the equation of the line
that is parallel to x − 3y = 15 through the
point (−3, 2) in slope-intercept form.
So, our slope is .
− =
− = − +
= −
x 3y 15
3y x 15
1
y x 5
3
1
3
( )− = +
= + +
= +
1
y 2 x 3
3
1
y x 1 2
3
1
y x 3
3

11. Example: Write an equation of the line that
is perpendicular to that goes
through the point (8, −3), in point-slope
form.
= −y 4x 3

12. Example: Write an equation of the line that
is perpendicular to that goes
through the point (8, −3), in point-slope
form.
orig. slope = ⊥ slope =
= −y 4x 3
−
1
4
4
1
( )+ = − −
1
y 3 x 8
4