Parallel lines have the same slope and never intersect. Two lines are parallel if their slopes are equal. The equation of a line parallel to a given line can be found using the point-slope formula with the same slope. Perpendicular lines intersect at a right angle. Two lines are perpendicular if the product of their slopes is -1. The equation of a line perpendicular to a given line can be found using the point-slope formula with the slope of the perpendicular line being the negative reciprocal of the given line's slope. Examples are provided of finding if lines are parallel or perpendicular and constructing parallel and perpendicular lines through a given point.

1. Parallel and Perpendicular
Lines

2. Parallel Lines
 Two lines with the same slope are said to be parallel
lines. If you graph them they will never intersect.
 We can decide algebraically if two lines are parallel by
finding the slope of each line and seeing if the slopes
are equal to each other.
 We can find the equation of a line parallel to a given line
and going through a given point by:
a.) first finding the slope m of the given line;
b.) finding the equation of the line through the given
point with slope m.

3. Testing if Lines are Parallel
Are the lines parallel?
12 3 9 and -8 2 14
x y x y
    
Find the slope of 12 3 9
3 12 9
4 3
x y
y x
y x
  
  
  
The slope m = -4
Find the slope of 8 2 14
2 8 14
4 7
x y
y x
y x
  
  
  
The slope m = -4
Since the slopes are equal the lines are parallel.

4. Graphs of Parallel Lines
The red line is the graph of
y = – 4x – 3
and the blue line is the graph of
y = – 4x – 7

5. Practice Testing if Lines are
Parallel
Are the lines 6 3 5 and 2 4 4
x y y x
     parallel?
6 3 5
3 6 5
5
2
3
2
x y
y x
y x
m
 
   
 

2 4 4
2 2
2
y x
y x
m
  
  
 
Since the slopes are different
the lines are not parallel.
Are the lines 2 4 and 2 4 12
x y x y
    parallel?
2 4
2 4
1 2
2
1
2
x y
y x
y x
m
 
   
 

2 4 12
4 2 12
1 3
2
1
2
x y
y x
y x
m
 
   
 

Since the slopes are equal
the lines are parallel.

6. Constructing Parallel Lines
Find the equation of a line going through the point (3, -5) and
parallel to 2 8
3
y x
  
Using the point-slope equation where the slope m = -2/3
and
the point is (3, -5) we get    
2
5 3
3
2
5 2
3
2 3
3
y x
y x
y x
    
   
  

7. Practice Constructing Parallel
Lines
Find the equation of the line going through the point (4,1) and
parallel to 3 7
y x
  
 
1 3 4
1 3 12
3 13
y x
y x
y x
   
   
  
Find the equation of the line going through the point (-2,7) and
parallel to 2 8
x y
 
 
 
 
7 2 2
7 2 2
7 2 4
2 3
y x
y x
y x
y x
    
   
   
  

8. Perpendicular Lines
 Perpendicular lines are lines that intersect in a right angle.
 We can decide algebraically if two lines are perpendicular by finding
the slope of each line and seeing if the slopes are negative
reciprocals of each other. This is equivalent to multiplying the two
slopes together and seeing if their product is –1.
 We can find the equation of a line perpendicular to a given line and
going through a given point by:
a.) first finding the slope m of the given line;
b.) finding the equation of the line through the given point with
slope = –1 /m.

9. Testing if Lines Are
Perpendicular
1
Are the lines 2 5 and 4 perpendicular?
2
x y y x
   
Find the slope of 2 5 2
2 5
x y m
y x
   
  
1 1
Find the slope of 4
2 2
y x m
  
Since the slopes are negative reciprocals of each other the lines
are perpendicular. 1
2 1
2
 
  
 
 

10. Graphs of Perpendicular Lines
The red line is the graph of
y = – 2x + 5
and the blue line is the
graph of
y = – 1/2 x +4

11. Practice Testing if Lines Are
Perpendicular
Are the lines 6 3 5 and 2 4 4 perpendicular?
x y y x
    
6 3 5
3 6 5
5
2
3
2
x y
y x
y x
m
 
   
 

2 4 4
2 2
2
y x
y x
m
  
  
 
Since the slopes are not
negative reciprocals of
each other (their product
is not -1) the lines are
not perpendicular
Are the lines 2 4 and 4 2 6 perpendicular?
x y x y
   
2 4
2 4
1 2
2
1
2
x y
y x
y x
m
 
   
 

4 2 6
2 4 6
2 3
2
x y
y x
y x
m
 
  
  
 
Since the slopes are
negative reciprocals of
each other (their
product is -1) the lines
are perpendicular.

12. Constructing Perpendicular
Lines
Find the equation of a line going through the point (3, -5) and
perpendicular to 2 8
3
y x
  
The slope of the perpendicular line will be m = 3/2 Using
the point-slope equation where the slope m = 3/2 and
the point is (3, -5) we get
   
3
5 3
2
3 9
5
2 2
3 19
2 2
y x
y x
y x
   
  
 

13. Practice Constructing
Perpendicular Lines
Find the equation of the line going through the point (4,1) and
perpendicular to 3 7
y x
  
Find the equation of the line going through the point (-2,7) and
perpendicular to 2 8
x y
 

14. Practice Constructing
Perpendicular Lines
Find the equation of the line going through the point (4,1) and
perpendicular to 3 7
y x
  
 
1
1 4
3
1 4
1
3 3
1 1
3 3
y x
y x
y x
  
  
 
Find the equation of the line going through the point (-2,7) and
perpendicular to 2 8
x y
 
 
 
 
1
7 2
2
1
7 2
2
1
7 1
2
1 8
2
y x
y x
y x
y x
   
  
  
 