Supplementary angles are a pair of angles, which when added measure 180 degrees. For example, the angles 140° and 40° are supplementary since adding them together we get 180 degrees. When we join two supplementary angles, we form a straight line.

Here, we will look at a more detailed definition of supplementary angles along with diagrams to illustrate the concepts. In addition, we will learn how to find these angles and we will look at some problems where we will apply what we have learned.

## What is a supplementary angle?

Supplementary angles are pairs of angles such that the sum of their angles is equal to 180 degrees. These angles always come in pairs, so one angle is the supplement of another angle.

Although the measure of an angle in a straight line measures 180 degrees, it is not considered a supplementary angle since it does not appear in pairs. Similarly, we cannot have three angles or more supplementary angles even though their sum can be equal to 180 degrees.

#### EXAMPLES

The following are some examples of supplementary angles:

- Two angles each measuring 90 degrees.
- Angles that measure 50 and 130 degrees.
- Angles that measure 1 and 179 degrees.

We can have several types of supplementary angles. For example, it is possible to have adjacent angles, non-adjacent angles, and right angles.

### Adjacent supplementary angles

Two supplementary angles with a common vertex and a common segment are called adjacent supplementary angles. An example of these angles is the following diagram, where the angles share the OB segment and also add up to 180 degrees.

### Non-adjacent supplementary angles

These angles have the characteristic of being supplementary, but not adjacent. This means that they do not share a vertex or a segment. In the following example, we can see that the angles do not have a common vertex or a common segment.

However, these angles are complementary since they add up to 180° and form a straight line when joined.

### Right supplementary angles

A supplementary angle can be formed by two right angles. Recall that the right angles have an angle of 90° as in the following diagram.

## How to find a supplementary angle?

When the sum of two pairs of angles equals 180 degrees, we call that pair of angles supplements to each other. Therefore, we know that the sum of two supplementary angles is 180 degrees and each of them is called a supplement to the other. This means that the supplement of an angle is found by subtracting that angle from 180 degrees.

In general terms, if we have the angle *x*°, its supplement is **(180- x)°**. For example, the supplement of the angle 75° is obtained by subtracting it from 180°. Therefore, its supplement is (180-75)° = 105°.

## Properties of supplementary angles

The following are some of the fundamental properties of supplementary angles:

- Two angles are supplementary if they add up to 180 degrees.
- Three or more angles cannot be supplementary even if they add up to 180 degrees.
- Supplementary angles can be adjacent or non-adjacent.
- When we join two supplementary angles, we form a straight line.
- If two angles are supplementary, each angle is called a “supplement” or “supplementary angle” of the other angle.

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## Supplementary angles – Examples with answers

The concepts learned about supplementary angles are applied to solve the following examples. Each example has its respective solution, where the process and reasoning used are detailed.

**EXAMPLE 1**

Determine whether the angles 132° and 48° are supplementary angles.

##### Solution

To find the complementary angle, we subtract the angle from 90°:

90° – 35° = 55°

The complementary angle of 35° is 55°.

**EXAMPLE 2**

If we have an angle of 57°, what is its supplementary angle?

##### Solution

We know that the angles in a triangle add up to 180°, so we can use this and form the following equation:

A + 90° + C = 180°

A + C = 90°

Therefore, we know that angles A and C are complementary since they must add up to 90°.

A + 40° = 90°

A = 90° – 40°

A = 50°

**EXAMPLE 3**

Find the supplementary angles that have a difference of 28°.

##### Solution

We can use *x* to represent the small angle. This means that the large angle will be (90-*x*)°. Therefore, we have:

(90°-*x*)-*x* = 24°

90°-2*x* = 24°

2*x* = 66°

*x* = 33°

⇒ 90°-*x* = 90°-33°

= 67°

The two complementary angles are 33° and 67°.

**EXAMPLE 4**

What is the supplementary angle of 2/3 of 120°?

##### Solution

We start by calculating the given angle:

90/3 = 30°

⇒ 90°-30° = 60°

Therefore, the complementary angle is 60°.

**EXAMPLE 5**

Find the angle that is 68° less than its supplement.

##### Solution

We can use *x* to represent the angle we want to find, so the complementary angle is (90°-*x*). The question tells us that the difference between the angle and its complement is equal to 38°. Therefore, we have:

(90-*x*)-*x* = 38

90-2*x* = 38

-2*x* = 38-90

-2*x* = -38

*x* = 19

The angle is 19°.

## Supplementary angles – Practice problems

Put into practice what you have learned about supplementary angles to solve the following problems. If you need help with this, you can look at the solved examples above.

## See also

Interested in learning more about angles? Take a look at these pages:

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