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8.2: Angle Pairs

Difficulty Level: At Grade Created by: CK-12
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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 \begin{align*}90^{\text{o}}.\end{align*}90o. 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 \begin{align*}180^{\text{o}}.\end{align*}180o. 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 \begin{align*}180^{\text{o}}.\end{align*}180o. Therefore these angles are supplementary angles.

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

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

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 \begin{align*}90^{\text{o}}\end{align*}90oangle in half, what type of angles will she form? What will be the value of those angles? 

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

Complementary angles

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

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

Then, write the size of each angle.

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

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

Example 2

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

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

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

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

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

Then, draw a conclusion.

Neither complementary nor supplementary.

The answer is neither.

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

Example 3

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

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

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

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

Then, draw a conclusion.

Neither complementary nor supplementary.

The answer is neither.

Example 4

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

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

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

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

Then, draw a conclusion.

The angles are complementary.

The answer is complementary angles.

Example 5

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

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.

The answer is supplementary angles.

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*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.2.

Resources

 

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Date Created:
Dec 02, 2015
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