The document explains how to determine parallel and perpendicular lines using their slopes. It includes methods for writing equations of lines that are parallel or perpendicular to given lines through specific points. Additionally, the document discusses different types of line interactions, such as inconsistent, consistent independent, and consistent dependent lines based on their slopes and y-intercepts.

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 LinesParallel LinesParallel LinesParallel Lines TheoremTheoremTheoremTheorem – 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
1 3 4
2
2 0 2
sm
− − −
= = =
− − −
3 1 4
2
1 1 2
tm
− − −
= = =
− − −

3. Perpendicular LinesPerpendicular LinesPerpendicular LinesPerpendicular Lines TheoremTheoremTheoremTheorem – 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
3 1 4
2
1 1 2
tm
− − −
= = =
− − −
0 1 1 1
3 0 3 3
qm
− −
= = =
− − −
1p qm m = − ⇒ ⊥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
:
3 ( 3) 6 3
AB m
− −
= = =
− −
2 4 6 3
:
0 ( 4) 4 2
CD m
− − −
= = = −
− −

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

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 .
3 15
3 15
1
5
3
x y
y x
y x
− =
− = − +
= −
1
3
( )
1
2 3
3
1
1 2
3
1
3
3
− = +
= + +
= +
y x
y x
y x

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

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 =
4 3y x= −
1
4
−
4
1
( )
1
3 8
4
+ = − −y x