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# 9.6: Supplementary and Complementary Angles

Difficulty Level: At Grade Created by: CK-12

Paul is fixing a house. The wood frames have already been placed and he is almost done with one of the rooms. He looks around the room and sees that along each wall, the boards are placed at a 90 degree angle. There are pairs of 90 degree angles next to each other. He knows that there is a special name for the angle pair, but he can't remember the name. What type of angle pair is formed by two 90 degree angles?

In this concept, you will learn about complementary and supplementary angles.

### Guidance

Sometimes, two angles are a part of each other or are connected to each other. These angles are called angle pairs.

Let's look at two special types of angle pairs, supplementary angles or complementary angles.

Supplementary angles are two angles whose sum is equal to 180\begin{align*}180^\circ\end{align*}. In other words when you add the measure of one angle in the pair with the other angle in the pair, they equal 180 degrees.

These two angles are supplementary because together they form a straight line. You can also tell that they are supplementary because when you add their angle measures the sum is equal to 180 degrees.

120+60=180\begin{align*}120 + 60 = 180^\circ\end{align*}

Complementary angles are a pair of angles whose sum is 90\begin{align*}90^\circ\end{align*}. Here is an example of complementary angles.

If you add up the two angle measures, the sum is equal to 90 degrees. Therefore, the two angles are complementary.

You can find missing angle measures by using information about supplementary and complementary angles.

Let's look at an example.

Find the measure of x\begin{align*}x\end{align*}.

First, identify that these two angles are supplementary. They form a straight line. The total number of degrees in a straight line is 180. Therefore, you can write the following equation to solve.

130+xx=180=50\begin{align*}130 + x & = 180 \\ x & = 50^\circ\end{align*}

The missing angle is equal to 50\begin{align*}50^\circ\end{align*}.

Here the two angles are complementary. Therefore the sum of the two angles is equal to 90 degrees. Write an equation and solve for the missing angle measure.

75+xx=90=15\begin{align*}75 + x & = 90 \\ x & = 15^\circ\end{align*}

The missing angle measure is equal to 50\begin{align*}50^\circ\end{align*}.

### Guided Practice

Identify the angle pair as complementary or supplementary.

First, add the two angles to calculate their sum.

75 + 105 = 180

Then, identify which angle pair adds up to 180 degrees.

Supplementary angles

### Examples

Write whether each pair is complementary or supplementary.

#### Example 1

If the sum of the angles is equal to 180 degrees.

First, identify which angle pair adds up to 180 degrees.

Supplementary angles

#### Example 2

If one angle is 60 degrees and the other angle is 120 degrees.

First, add the two angles together to find the sum.

60 + 120 = 180

Then, identify which angle pair adds up to 180 degrees.

Supplementary angles

#### Example 3

First, add the two angles to calculate their sum.

30 + 60 = 90

Then, identify which angle pair adds up to 90 degrees.

Complementary angles

The answer is that the pair are complementary angles.

Remember Paul and the house? He has an angle pair that is formed by two 90 degree angles. What type of angle pair is formed when two 90 degree angles are combined?

First,add the two angles to calculate their sum.

90 + 90 = 180

Then, identify which angle pair adds up to 90 degrees.

Supplementary angles

The answer is that the pair are supplementary angles.

### Explore More

Identify each angle pair as supplementary or complementary angles.

1.

2.

3.

4.

Use what you have learned about complementary and supplementary angles to answer the following questions.

5. If two angles are complementary, then their sum is equal to _________ degrees.

6. If two angles are supplementary, then their sum is equal to ________ degrees.

7. True or false. If one angle is 120\begin{align*}120^\circ\end{align*}, then the second angle must be equal to 90\begin{align*}90^\circ\end{align*} for the angles to be supplementary.

8. True or false. If the angles are supplementary and one angle is equal to 100\begin{align*}100^\circ\end{align*}, then the other angle must be equal to 80\begin{align*}80^\circ\end{align*}.

9. True or false. The sum of complementary angles is 180\begin{align*}180^\circ\end{align*}.

10. True or false. The sum of supplementary angles is 90\begin{align*}90^\circ\end{align*}.

Identify whether the angles are supplementary, complementary or neither based on the angle measures.

11. One angle is 50 degrees. The other angle is 130 degrees.

12. One angle is 30 degrees. The other angle is 60 degrees.

13. One angle is 112 degrees. The other angle is 70 degrees.

14. One angle is 110 degrees. The other angle is 50 degrees.

15. One angle is 35 degrees. The other angle is 55 degrees.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.6.

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