Perpendicular Lines – Definition, Properties and Examples

Perpendicular lines are two lines that intersect each other at an angle of 90°. We can also say that, if two lines are perpendicular, their intersection forms a right angle. Here, we will learn more details of the perpendicular lines.

We will learn about its properties and solve some exercises.

GEOMETRY
diagram of perpendicular lines 1

Relevant for

Learning about the definition and properties of perpendicular lines.

See definition

GEOMETRY
diagram of perpendicular lines 1

Relevant for

Learning about the definition and properties of perpendicular lines.

See definition

What are perpendicular lines?

Perpendicular lines are two straight lines that are characterized by forming an angle of 90° with each other. The 90° angle is also referred to as a right angle and can be represented using a small square as shown in the diagram below.

diagram of perpendicular lines 1

These lines intersect at an angle of 90° and are therefore perpendicular. Two lines must intersect and form a 90° angle to be considered perpendicular.


Properties of perpendicular lines

Perpendicular lines can be identified because they form an L-shaped intersection. The corresponding angle formed at the intersection vertex is equal to 90°. Lines have to intersect to form perpendicular lines, but not all intersecting lines are perpendicular. The following are the properties of perpendicular lines:

  • Perpendicular lines always intersect each other.
  • The angle formed between two perpendicular lines is always 90°.
  • The slopes of the perpendicular lines are reciprocal and negative of each other.
  • 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 perpendicular?

We can determine if two lines are perpendicular using two main methods.

Using the angles

For two lines to be perpendicular, their angle at the point of intersection must be equal to 90°. The 90° angle is represented by a small square. Therefore, if we know the angle formed, we can easily determine whether or not two lines are perpendicular.

Using the slopes

If we know the equations of the lines or can derive them, we can use their slopes to determine whether the lines are perpendicular. Two lines are perpendicular if their slopes are the negative reciprocal of each other. That is, we have:

$latex m_{1} = – \frac{1}{m_{2}}$

where, $latex m_{1}, ~ m_{2}$ are the slopes of the lines.


Differences between perpendicular and parallel lines

Two lines are parallel when they do not cross each other no matter how long they are extended. Parallel lines always remain equidistant. In the following diagram, we can look at the difference between parallel lines and perpendicular lines. Lines AB and CD are perpendicular, while lines EF and GH are parallel.

difference between perpendicular and parallel lines

Parallel lines are represented by the symbol ||. For example, EF||GH indicates that lines EF and GH are parallel. On the other hand, the symbol used to represent two perpendicular lines is ⊥. For example, AB⊥CD indicates that lines AB and CD are perpendicular.


Solved problems of perpendicular lines

The following are some exercises for applying perpendicular lines.

EXAMPLE 1

The following lines in the diagram are perpendicular. What are the measures of all the angles formed?

diagram of perpendicular lines 1

Solution: The angles formed by two perpendicular lines are always right angles. Therefore, all the angles formed are equal to 90°.

EXAMPLE 2

In the diagram below, line AB is perpendicular to line CD. Find the value of the angle x.

exercise with perpendicular lines

Solution: The perpendicular lines form 90° angles at the point of intersection. This means that the sum of angles x and 53° must equal 90°. Thus, we have:

53°+x = 90°

x = 90°-53°

x = 37°

Angle x measures 37°.


See also

Interested in learning more about perpendicular and parallel lines? Take a look at these pages:

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