Parallel and perpendicular lines – Formulas and Examples

Parallel lines are lines that have the same slope. These lines never cross each other, no matter how much they are extended. On the other hand, perpendicular lines are two lines that intersect at a 90° angle with each other. The slopes of perpendicular lines are negative reciprocals of each other.

Here, we will learn everything related to parallel and perpendicular lines. We will look at their definitions, some important characteristics, and some examples.

GEOMETRY
Parallel and perpendicular lines

Relevant for

Learning about parallel and perpendicular lines.

See definitions

GEOMETRY
Parallel and perpendicular lines

Relevant for

Learning about parallel and perpendicular lines.

See definitions

Definitions of parallel lines and perpendicular lines

What are parallel lines?

Parallel lines are two or more lines that lie in the same coordinate plane and never intersect with each other. No matter how long we extend two parallel lines, they will never touch.

The following is a diagram with two pairs of parallel lines.

diagram of parallel lines 1

The main characteristic of two or more parallel lines is that they have the same slope. That is, the angle of inclination of the parallel lines is the same.

What are perpendicular lines?

Perpendicular lines are two lines that intersect each other at a 90° angle. That is, perpendicular lines form a right angle at their point of intersection.

The following is a diagram of two perpendicular lines.

diagram of perpendicular lines 1

Properties of parallel and perpendicular lines

Parallel lines can be identified because they have the same slope and never meet. On the other hand, perpendicular lines can be identified because they form an L-shaped intersection.

The following are some important properties of parallel lines:

  • Parallel lines never intersect with each other.
  • The distance between two parallel lines always remains constant.
  • The slopes of the parallel lines are the same.

The following are some important properties of perpendicular lines:

  • Perpendicular lines always intersect one another.
  • The angle of intersection between two perpendicular lines is always 90°.
  • The slopes of perpendicular lines are reciprocals and negatives of each other. That is, we have $latex m_{1}=-\frac{1}{m_{2}}$.
  • If a line is perpendicular to a line that is parallel to other lines, then the line is perpendicular to all other lines.


How to determine if two lines are parallel or perpendicular?

We can determine if two lines are parallel or perpendicular by using the angles formed between the lines or by using their slopes.

Using the angles formed

For two lines to be perpendicular, the angle of intersection must be equal to 90°. On the other hand, for the lines to be parallel, the angle formed by the lines with respect to the horizontal must be equal.

Using the slopes of the lines

For two lines to be parallel, their slopes must be the same. On the other hand, for two lines to be perpendicular, their lines must be negative reciprocals of each other.

That is, for two lines that have slopes $latex m_{1}$ and $latex m_{2}$, we have:

Parallel: $latex m_{1}=m_{2}$

Perpendicular: $latex m_{1}=-\frac{1}{m_{2}}$

Remember that in the equation of a line $latex y=mx+b$, m is the slope.

In addition, if we know two points on the line, its slope is $latex m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$, where, $latex (x_{1},~y_{1})$ and $latex (x_{2},~y_{2})$ are the coordinates of the points.


Examples of parallel and perpendicular lines

The following are some examples on the application of parallel and perpendicular lines.

EXAMPLE 1

If we have the line $latex 8x-2y=10$, what is the slope of a parallel line?

Solution: For two lines to be parallel, they must have the same slope. Therefore, we have to find the slope of the given line.

For this, we have to write the line in the form $latex y=mx+b$. This means that we solve for y:

$latex 8x-2y=10$

$latex -2y=10-8x$

$latex y=4x-5$

The slope of the line is $latex m=4$. Therefore, the slope of the parallel line must also be $latex m=4$.

EXAMPLE 2

In the diagram below, the lines $latex l_{1}$ and $latex l_{2}$ are parallel. In addition, we also have the angle ∠a=25°. Find the measure of angle b.

diagram of parallel lines exercise

Solution: The lines are parallel, so their angle of inclination is the same. This means that the line that cuts both parallel lines must form the same angles in both lines.

Therefore, we can move the angle a, from the first line to the second line. Therefore, angles a and b are supplementary, that is, they add up to 180°. Thus, we have:

∠b=180°-25°=155°

Angle b is equal to 155°.

EXAMPLE 3

Line AB in the diagram below is perpendicular to line CD. Determine the value of angle x.

exercise with perpendicular lines

Solution: Perpendicular lines form 90° angles at the point of intersection. This means that angles x and 53° must add up to 90°. Forming an equation and solving, we have:

$latex 53^{\circ}+x=90^{\circ}$

$latex x=90^{\circ}-53^{\circ}$

$latex x=37^{\circ}$

Angle x is equal to 37°.

EXAMPLE 4

If a line has the equation $latex 6x+3y=12$, find the slope of a perpendicular line.

Solution: We have to start by finding the slope of the given line. For this, we write the line in the form $latex y=mx+b$:

$latex 6x+3y=12$

$latex 3y=-6x+12$

$latex y=-2x+4$

We see that the slope is $latex m_{1}=-2$.

Now, to find the slope of the perpendicular line, we recall that perpendicular lines have slopes that are negative reciprocals of each other.

Therefore, the slope of the perpendicular line is $latex m_{2}=\frac{1}{2}$


See also

Interested in learning more about angles and lines? You can take a look at these pages:

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