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# 8.3: Supplementary and Complementary Angle Pairs

Difficulty Level: At Grade Created by: CK-12
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Have you ever tried to figure out an angle measure? Look at what happened at the art museum.

Justin was looking at a painting with two intersecting lines on it. One of the lines formed a straight line and the other intersected with the first line.

"What do you think the measure is of the smaller angle?" he asked Susan who was standing nearby.

"I think it is about 30\begin{align*}30^\circ\end{align*}," Susan said.

"That's exactly what I was thinking," Justin added.

If Susan and Justin are correct, can you figure out the other missing angle?

This Concept will show you how reasoning can help you figure out the measures of missing angles.

### Guidance

As we have seen, we identify complementary and supplementary angles by their sum. This means that we can also find the measure of one angle in a pair if we know the measure of the other angle. For instance, because we know that complementary angles always add up to 90\begin{align*}90^\circ\end{align*}, we can calculate the measurement of one angle in a pair of complementary angles. Let’s see how this works.

We can see that together, C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*} form a right angle. Therefore they are complementary, and they add up to 90\begin{align*}90^\circ\end{align*}. We know that C\begin{align*}C\end{align*} has a measure of 44\begin{align*}44^\circ\end{align*}. How can we find the measure of angle D\begin{align*}D\end{align*}?

To find the measurement of angle D\begin{align*}D\end{align*}, we simply subtract the measure of angle C\begin{align*}C\end{align*} from 90.

C+D44+DDD=90=90=9044=46

In order for these two angles to be complementary, as the problem states, they must add up to 90\begin{align*}90^\circ\end{align*}. Angle D\begin{align*}D\end{align*} therefore measures 46\begin{align*}46^\circ\end{align*}. We can check our calculation by adding angles C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*}. Their sum must be equal to 90\begin{align*}90^\circ\end{align*}.

44+46=90\begin{align*}44^\circ + 46^\circ = 90^\circ\end{align*}

We can follow the same process to find the unknown angle in a pair of supplementary angles. As with complementary angles, if we know the measure of one angle in the pair, we can find the measure of the other.

Angles P\begin{align*}P\end{align*} and Q\begin{align*}Q\end{align*} are supplementary angles. If angle P\begin{align*}P\end{align*} measures 112\begin{align*}112^\circ\end{align*}, what is the measure of angle Q\begin{align*}Q\end{align*}?

We know that supplementary angles have a total of 180\begin{align*}180^\circ\end{align*} Therefore we can subtract the measurement of the angle we know, angle P\begin{align*}P\end{align*}, from 180\begin{align*}180^\circ\end{align*} to find the measure of angle Q\begin{align*}Q\end{align*}.

P+Q112+QQQ=180=180=180112=68

Angle Q\begin{align*}Q\end{align*} is 68\begin{align*}68^\circ\end{align*}. We can check our calculation by adding angles P\begin{align*}P\end{align*} and Q\begin{align*}Q\end{align*}. Remember, in order to be supplementary angles, their sum must be equal to 180\begin{align*}180^\circ\end{align*}.

68+112=180\begin{align*}68^\circ + 112^\circ = 180^\circ\end{align*}

We can call this finding the complement or the supplement.

Armed with our knowledge of complementary and supplementary angles, we can often find the measure of unknown angles. We can use logical reasoning to interpret the information we have been given in order to find the unknown measure. Take a look at the diagram below.

Can we find the measure of angle X\begin{align*}X\end{align*}? We can, if we apply what we have learned about supplementary angles. We know that supplementary angles add up to 180\begin{align*}180^\circ\end{align*}, and that 180\begin{align*}180^\circ\end{align*} is a straight line. Look at the diagram. The 80\begin{align*}80^\circ\end{align*} angle and angle X\begin{align*}X\end{align*} together form a straight line, so we can deduce that they are supplementary angles. That means we can set up an equation to solve for X\begin{align*}X\end{align*}.

80+x=180\begin{align*}80 + x = 180\end{align*}

The equation shows what we already know: the sum of supplementary angles is 180\begin{align*}180^\circ\end{align*}. We can find the measure of the unknown angle by solving for X\begin{align*}X\end{align*}.

80+xxx=180=18080=100

The measure of the unknown angle in this supplementary pair is 100\begin{align*}100^\circ\end{align*}.

We can check our work by putting this value in for X\begin{align*}X\end{align*} in the equation.

80+100=180\begin{align*}80 + 100 = 180\end{align*}

Now it's time for you to apply what you have learned. Find the complement or supplement in each example.

#### Example A

Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 33\begin{align*}33^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

Solution:57\begin{align*}57^\circ\end{align*}

#### Example B

Angles C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*} are supplementary. Angle C\begin{align*}C\end{align*} is 59\begin{align*}59^\circ\end{align*}. Find the measure of angle D\begin{align*}D\end{align*}.

Solution:121\begin{align*}121^\circ\end{align*}

#### Example C

Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are supplementary. Angle A\begin{align*}A\end{align*} is 169\begin{align*}169^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

Solution:11\begin{align*}11^\circ\end{align*}

Here is the original problem once again.

Justin was looking at a painting with two intersecting lines on it. One of the lines formed a straight line and the other intersected with the first line.

"What do you think the measure is of the smaller angle?" he asked Susan who was standing nearby.

"I think it is about 30\begin{align*}30^\circ\end{align*}," Susan said.

"That's exactly what I was thinking," Justin added.

If Susan and Justin are correct, can you figure out the other missing angle?

To figure this out, we can use reasoning and the dilemma to hunt for clues. First, notice that the painting had one straight line. We know that the measure of a straight line is 180circ\begin{align*}180^circ\end{align*}. Given this, we can write an equation.

x+30=180\begin{align*}x + 30 = 180\end{align*}

The 30 is the measure of the angle that Justin and Susan figure out.

Now we can solve for the unknown variable.

x=150\begin{align*}x = 150^\circ\end{align*}

This is our answer.

### Vocabulary

Here are the vocabulary words in this Concept.

Acute Angle
an angle whose measure is less than 90\begin{align*}90^\circ\end{align*}
Obtuse Angle
an angle whose measure is greater than 90\begin{align*}90^\circ\end{align*}
Right Angle
an angle whose measure is equal to 90\begin{align*}90^\circ\end{align*}
Straight Angle
an angle whose measure is equal to 180\begin{align*}180^\circ\end{align*}
Degrees
how an angle is measured
Angle Pairs
when the measures of two angles are added together to form a special relationship
Supplementary Angles
angle pairs whose sum is 180\begin{align*}180^\circ\end{align*}
Complementary Angles
angle pairs whose sum is 90\begin{align*}90^\circ\end{align*}

### Guided Practice

Here is one for you to try on your own.

What is the measure of angle R\begin{align*}R\end{align*}?

Answer

How can we use what we have learned to find the measure of angle R\begin{align*}R\end{align*}? Can we determine whether the two angles have a relationship with each other? Together, they form a right angle. They must be a pair of complementary angles, so we know their sum is 90\begin{align*}90^\circ\end{align*}. Again, we can set up an equation to solve for R\begin{align*}R\end{align*}, the unknown angle.

R+22=90\begin{align*}R + 22 = 90\end{align*}

This equation represents what we know, that the sum of these two complementary angles is 90\begin{align*}90^\circ\end{align*}. Now we solve for R\begin{align*}R\end{align*}.

R+22RR=90=9022=68

The measure of the unknown angle is 68\begin{align*}68^\circ\end{align*}. We can check our answer by putting this value in for R\begin{align*}R\end{align*} in the equation.

68+22=90\begin{align*}68 + 22 = 90^\circ\end{align*}

### Video Review

Here is a video for review.

### Practice

Directions: Find the measure of missing angle for each pair of complementary or supplementary angles.

1. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 63\begin{align*}63^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

2. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 83\begin{align*}83^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

3. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 3\begin{align*}3^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

4. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 23\begin{align*}23^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

5. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 70\begin{align*}70^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

6. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle A\begin{align*}A\end{align*} is 29\begin{align*}29^\circ\end{align*}. Find the measure of angle B\begin{align*}B\end{align*}.

7. Angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}66^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

8. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complementary. Angle \begin{align*}A\end{align*} is \begin{align*}87^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

9. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}33^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

10. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}103^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

11. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}73^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

12. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}78^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

13. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}99^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

14. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}110^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

15. Angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are supplementary. Angle \begin{align*}A\end{align*} is \begin{align*}127^\circ\end{align*}. Find the measure of angle \begin{align*}B\end{align*}.

### Vocabulary Language: English

Acute Angle

Acute Angle

An acute angle is an angle with a measure of less than 90 degrees.
Complementary angles

Complementary angles

Complementary angles are a pair of angles with a sum of $90^{\circ}$.
Obtuse angle

Obtuse angle

An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.
Straight angle

Straight angle

A straight angle is a straight line equal to $180^{\circ}$.
Supplementary angles

Supplementary angles

Supplementary angles are two angles whose sum is 180 degrees.

At Grade

Oct 29, 2012

## Last Modified:

Jul 08, 2015
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