1. If and are slopes of parallel lines, then .
Today we will explore the Essential Question, “How are the slopes of parallel lines and the slopes of perpendicular lines related?”
Parallel lines are lines that lie in the same plane but do not intersect. The graph shows a pair of lines that are parallel to each other. x y 2 4 6 8 10 -2 -4 -6 -8
-10 1
Let's choose 2 points on each of the lines and determine the slope of each line.
Moving from left to right on line 1, the rise between the two points is +9 and the run is +9. +9 +6 +9 +6
Moving from left to right on line 2, the rise between the two points is +6 and the run is +6.
If and are slopes of parallel lines, then
For example:
The lines defined by the equations and are parallel lines because both lines have a slope of 2 and the lines have different y-intercepts.
Note: If two lines have the same slope and the same y-intercept, then the lines are the same and are not considered parallel.

2. Numbers such as and +3 are called opposite reciprocals.
Perpendicular lines are lines that intersect to form a right angle as shown in the diagram. x y 2 4 6 8 10 -2 -4 -6 -8
-10
On line 1, 2
On line 2, 1
Numbers such as and +3 are called opposite reciprocals. -2 +6 +2
“Opposite" means that the numbers have opposite signs, one is negative and one is positive. +6
"Reciprocal" means that the numbers are reciprocals of each other.
Other examples of "opposite reciprocals" are:
If and are slopes of perpendicular lines, then is the opposite reciprocal of
For example:
The lines defined by the equations and are perpendicular lines because their slopes are 2 and which are opposite reciprocals of each other.

3. Modeled Examples:
Example 1: Are the lines whose equations are given below, parallel, perpendicular, or neither?
Solution:
Each equation is written in the form , where m is the slope and b is the y-intercept.
In both equations m = 5 .
This means that both lines have a slope of 5.
If two lines have the same slope and different y-intercepts, then the lines are parallel.
The y-intercepts of the lines are 4 and -4.
Therefore, the lines whose equations are given are parallel lines.

4. Example 2: Write the equation in slope-intercept form of a line that is perpendicular to and has a y-intercept of 4.
The formula for slope-intercept form is where m is the slope and b is the y-intercept.
The given equation is written in slope-intercept form as shown: so
A line perpendicular to the given line has a slope that is the opposite reciprocal of which is
Since the y-intercept of our line is 4, b = 4. 4
Use the formula and substitute as shown to obtain the equation of the line: m b

5. Guided Practice Problems:
1. Are the lines whose equations are given parallel, perpendicular, or neither?
Solution:
Each equation is written in the form , where m is the slope and b is the y-intercept.
In the top equation, m =
In the bottom equation m =
The slopes, and , are opposite reciprocals.
Therefore, the lines whose equations are given are perpendicular lines.

6. 2. Are the lines whose equations are given parallel, perpendicular, or neither?
Solution:
Each equation is written in the form , where m is the slope and b is the y-intercept.
In the top equation, m =
In the bottom equation m = 4.
The slopes, and , are reciprocals but not opposite reciprocals. Also the slopes are not equal.
Therefore, the lines whose equations are given are neither parallel nor perpendicular lines.

7. Write the equation in slope-intercept form of a line that is parallel to
and has a y-intercept of 8.
The formula for slope-intercept form is where m is the slope and b is the y-intercept.
The given equation is written in slope-intercept form as shown: so
A line parallel to the given line has a slope that is equal to the slope of the given line, which is
Since the y-intercept of our line is 8, b = 8. 8
Use the formula and substitute as shown to obtain the equation of the line: m b

8. 4. Write the equation in slope-intercept form of a line that is perpendicular to and has a y-intercept of
The formula for slope-intercept form is where m is the slope and b is the y-intercept.
The given equation is written in slope-intercept form as shown: so
A line perpendicular to the given line has a slope that is the opposite reciprocal of which is
Since the y-intercept of our line is ,
Use the formula and substitute as shown to obtain the equation of the line: m b or

9. Independent Practice Problems:
1. Are the lines whose equations are given parallel, perpendicular, or neither?
The lines are parallel because they have the same slope and different y-intercepts.
2. Write the equation in slope-intercept form of a line that is perpendicular to
and has a y-intercept of 0.
m = -2
The opposite reciprocal of -2 is
To write the equation of the line perpendicular to the given line, use
and b = 0.
Substitute into to obtain