A detailed tutorial on how to prove that the slopes of two parallel lines are equal. Questions on parallel lines are also included with their detailed solutions.

- 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.

__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

Tutorial on Equation of Line

Line Problems

Equations of Line Through Two Points And Parallel and Perpendicular.