
Prove that two parallel lines have equal slopes
 The figure below shows two parallel lines L1 and L2. Points A and B are on the line L1 and points C and D are on the line L2.
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 Let BM and DN be parallel to the y axis and AM and CN parallel to the x axis. Triangles ABM and CDN have all their corresponding sides parallel are therefore similar. Hence
BM / DN = MA / NC
 The above may be written
BM / MA = DN / NC
 According to the definition of slope, BM / MA is the slope of line L1 and DN / NC is the slope of L2 and are therefore equal.
Questions on parallel lines
Question 1
Which of the lines given by the equations
a) y = 2 x  3 b) 2 x  y = 2 c)  4 y + 2 x = 0 d)  4 y + 8 x = 9
are parallel?
Solution to Question 1
We first need to write each line in slope intercept form y = m x + b and find its slope m.
a) y = 2 x  3, slope = 2
b) 2 x  y = 2, solve for y, y = 2 x  2, slope = 2
c)  4 y + 2 x = 0, solve for y, y = (1 / 2) x, slope = 1 / 2
d)  4 y + 8 x = 9, solve for y, y = 2 x  9 / 4, slope = 2
The lines with equal slopes are the lines given in parts a) b) and d) and they are therefore parallel.
Question 2
Find k so that the lines with equations 6 k x  3 y = 9 and  4 x + 5 y = 7 are parallel.
Solution to Question 2
Solve for y and find the slope of each line
6 k x  3 y = 9 , solve for y, y = 2 k x  3 , slope = 2 k
 4 x + 5 y = 7 , solve for y , y = (4 / 5) x + 7 / 5 , slope = 4 / 5
The two lines are parallel and therefore their slopes are equal; hence
2 k = 4 / 5
Solve for k
k = 2 / 5
Question 3
Find a so that the line through the points (1 , 2) and (0 , 3) and the line through the points (a , 2) and ( 2 , 7) are parallel.
Solution to Question 3
Find the slope m_{1} through the points (1 , 2) and (0 , 3)
m_{1} = (3  2) /(0  1) =  1
Find the slope m_{2} through the points (a , 2) and (2 , 7)
m_{2} = (7  2) / (2  a) = 5 / (2  a)
The two lines are parallel, hence their slopes are equal
5 / (2  a) =  1
Solve for a
a = 3
Question 4
Find the coordinate of point D so that the points A(0 , 2) , B(2,6), C(8 , 8) and D are the vertices of a parallelogram.
Solution to Question 4
Let (a , b) be the coordinates of point D. For the points A, B, C and D to be the vertices of a parallelogram, segment AB must be parallel to segment DC and segment BC must be parallel to segment AD as shown in the figure below.
slope of segment AB = (6  2) / (2  0) = 2
slope of segment DC = (8  b) / (8  a)
for the segments AB and DC to be parallel, their slopes must be equal, hence the equation
2 = (8  b) / (8  a)
cross multiply to obtain
8  b = 16  2 a
slope of segment BC = (8  6) / (8  2) = 1 / 3
slope of segment AD = (b  2) / (a  0) = (b  2) / a
For the segments BC and AD to be parallel, their slopes must be equal
1 / 3 = (b  2) / a
cross multiply to obtain
a = 3 b  6
We now solve the system of the two equations in a and b obtained above and given by
8  b = 16  2 a
a = 3 b  6
Substitute a by 3 b  6 in the equation 8  b = 16  2 a to obtain
8  b = 16  2 (3 b  6)
Solve the above equation for b to obtain
b = 4
then use any of the two equations to find a
a = 6
The coordinates of point D are
D(6 , 4)
Question 5
Find the equation of a line L_{1} that passes through the point (2 , 1) and is parallel to the line L_{2} through the points (1 , 2) and (3 , 4).
Solution to Question 5
The slope m_{2} of the line L_{2} through the points (1 , 2) and (3 , 4) is given by
m_{2} = (4  2) / (3  (1) ) = 2 / 4 = 1 / 2
Let m_{1} be the slope of the line L_{1}. L_{1} and L_{2} being parallel their slopes are equal. Hence
m_{1} = m_{2} = 1 / 2
The equation of line L_{1} in point (2 , 1) slope m_{1} form is given by
y  1 = (1 / 2)(x  2)
which simplifies to
y = (1 / 2) x
Question 6
Find the equation of a line L_{1} that passes through the point (1 , 1) and is parallel to the line L_{2} with equation 2 y + 4 x = 2.
Solution to Question 6
The slope m_{2} of the line L_{2} is found by first writing its equation in slope intercept form y = m x + b. Hence solve 2 y + 4 x = 2 for y to obtain
y =  2 x + 1
The slope of L_{2} is
m_{2} =  2
Since L_{1} and L_{2} are parallel their slopes are equal. Hence
m_{1} = m_{2} =  2
The equation of line L_{1} in point (1 , 1) slope m_{1} form is given by
y  1 =  2(x  (1))
which simplifies to
y =  2 x  1
More References and Links
Equations of Lines in Different Forms,
Tutorial on Equation of Line
Line Problems
Equations of Line Through Two Points And Parallel and Perpendicular.
