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# 8.2: Angle Pairs

Difficulty Level: At Grade Created by: CK-12
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Practice Angle Pairs

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Lily wants to change the hopscotch design so that each player's foot can only land in half of each box. Her plan is to draw a line from one corner of each box to the diagonal corner of each box. She takes out a protractor to measure half of a corner angle in Box 1. What should be the measure of half of a corner angle?

In this concept, you will learn how to identify and use angle pairs.

### Identifying and Using Angle Pairs

Two angles together are angle pairs. Sometimes, the measures of these angles form a special relationship. One special relationship is called complementary angles. Complementary angles are angle pairs that add up to exactly 90o.\begin{align*}90^{\text{o}}.\end{align*} In other words, when put together, the two angles form a right angle.

Below are some pairs of complementary angles.

Supplementary angles are two angles that add up to exactly 180o.\begin{align*}180^{\text{o}}.\end{align*} When put together, the two angles form a straight angle. Take a look at the pairs of supplementary angles below.

Let’s look at two examples.

Classify the following pairs of angles as either complementary or supplementary.

The sum of the angles in Figure 1 is 180o.\begin{align*}180^{\text{o}}.\end{align*} Therefore these angles are supplementary angles.

The sum of the angles in Figure 2 is 90o.\begin{align*}90^{\text{o}}.\end{align*} Therefore these angles are complementary angles.

Remember, complementary angles add up to 90o,\begin{align*}90^{\text{o}},\end{align*} and supplementary angles add up to 180o.\begin{align*}180^{\text{o}}.\end{align*}

### Examples

#### Example 1

Earlier, you were given a problem about Lily, who wants to divide spaces in half by drawing a line from one corner to the diagonal corner.

Lily needs to figure out the value of half of a corner angle. If she divides a 90o\begin{align*}90^{\text{o}}\end{align*}angle in half, what type of angles will she form? What will be the value of those angles?

First, what type of angle pair equals 90o?\begin{align*}90^{\text{o}}?\end{align*}

Complementary angles

Next, what is the value of half of 90o\begin{align*}90^{\text{o}}\end{align*}

45o\begin{align*}45^{\text{o}}\end{align*}

Then, write the size of each angle.

45o, 45o\begin{align*}45^{\text{o}} \! , \ 45^{\text{o}}\end{align*}

The answer is that Lily will create complementary angles that each have a measure of 45o.\begin{align*}45^{\text{o}}.\end{align*}

#### Example 2

Solve the following problem by identifying the angle pair as complementary, supplementary or neither.

Angle A=36o,\begin{align*}A = 36^{\text{o}} \! ,\end{align*} Angle B=45o.\begin{align*}B = 45^{\text{o}}.\end{align*}

First, add the values of the angles.
36o+45o=81o\begin{align*}36^{\text{o}} + 45^{\text{o}} = 81^{\text{o}}\end{align*}

Next, check to see if the sum is 90o\begin{align*}90^{\text{o}}\end{align*} or 180o\begin{align*}180^{\text{o}}\end{align*}.

The sum is 81o.\begin{align*}81^{\text{o}}.\end{align*}

Then, draw a conclusion.

Neither complementary nor supplementary.

Identify the following angle pairs as complementary, supplementary or neither.

#### Example 3

Angle A=23o,\begin{align*}A = 23^{\text{o}},\end{align*} Angle \begin{align*}B = 45^{\text{o}}\end{align*}

First, add the values of the angles.
\begin{align*}23^{\text{o}} + 45^{\text{o}} = 68^{\text{o}}\end{align*}

Next, check to see if the sum is \begin{align*}90^{\text{o}}\end{align*} or \begin{align*}180^{\text{o}}.\end{align*}

The sum is \begin{align*}68^o\end{align*}

Then, draw a conclusion.

Neither complementary nor supplementary.

#### Example 4

Angle \begin{align*}A = 45^{\text{o}} \!,\end{align*} Angle \begin{align*}B = 45^{\text{o}}\!.\end{align*}

First, add the values of the angles.
\begin{align*}45^{\text{o}} + 45^{\text{o}} = 90^{\text{o}}\end{align*}

Next, check to see if the sum is \begin{align*}90^{\text{o}}\end{align*} or \begin{align*}180^{\text{o}} \!.\end{align*}

The sum is \begin{align*}90^{\text{o}} \!.\end{align*}

Then, draw a conclusion.

The angles are complementary.

#### Example 5

Angle \begin{align*}A = 103^{\text{o}} \!,\end{align*} Angle \begin{align*}B = 77^{\text{o}}\end{align*}

First, add the values of the angles.
\begin{align*}103^o + 77^o = 180^o\end{align*}

Next, check to see if the sum is \begin{align*}90^{\text{o}}\end{align*} or \begin{align*}180^{\text{o}}\!.\end{align*}

The sum is \begin{align*}180^{\text{o}}\!.\end{align*}

Then, draw a conclusion.

The angles are supplementary.

### Review

Identify whether the pairs below are complementary, supplementary or neither.

1. An angle pair whose sum is \begin{align*}180^{\text{o}}\end{align*}
2. Angle \begin{align*}A = 90^{\text{o}}\end{align*} Angle \begin{align*}B\end{align*} is \begin{align*}45^{\text{o}}\end{align*}
3. Angle \begin{align*}C = 125^{\text{o}}\end{align*} Angle \begin{align*}B = 55^{\text{o}}\end{align*}
4. An angle pair whose sum is \begin{align*}180^{\text{o}}\end{align*}
5. An angle pair whose sum is \begin{align*}245^{\text{o}}\end{align*}
6. An angle pair whose sum is \begin{align*}80^{\text{o}}\end{align*}
7. An angle pair whose sum is \begin{align*}90^{\text{o}}\end{align*}
8. An angle pair whose sum is \begin{align*}55^{\text{o}}\end{align*}
9. An angle pair whose sum is \begin{align*}120^{\text{o}}\end{align*}
10. An angle pair whose sum is \begin{align*}95^{\text{o}}\end{align*}
11. An angle pair whose sum is \begin{align*}201^{\text{o}}\end{align*}
12. An angle pair whose sum is \begin{align*}190^{\text{o}}\end{align*}

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