# 1.5: Angle Pairs

**At Grade**Created by: CK-12

## Learning Objectives

- Recognize complementary angles supplementary angles, linear pairs, and vertical angles.
- Apply the Linear Pair Postulate and the Vertical Angles Theorem.

## Review Queue

- Find .
- Find .
- Find .

**Know What?** A compass (as seen to the right) is used to determine the direction a person is traveling. The angles between each direction are very important because they enable someone to be more specific with their direction. A direction of , would be straight out along that northwest line.

What headings have the same angle measure? What is the angle measure between each compass line?

## Complementary Angles

**Complementary:** Two angles that add up to .

Complementary angles **do not** have to be:

- congruent
- next to each other

**Example 1:** The two angles below are complementary. . What is ?

**Solution:** Because the two angles are complementary, they add up to . Make an equation.

**Example 2:** The two angles below are complementary. Find the measure of each angle.

**Solution:** The two angles add up to . Make an equation.

However, you need to find each angle. Plug back into each expression.

## Supplementary Angles

**Supplementary:** Two angles that add up to .

Supplementary angles **do not** have to be:

- congruent
- next to each other

**Example 3:** The two angles below are supplementary. If what is ?

**Solution:** Set up an equation. However, instead of equaling , now it is .

**Example 4:** What is the measure of two congruent, supplementary angles?

**Solution:** Supplementary angles add up to . Congruent angles have the same measure. So, , which means two congruent, supplementary angles are right angles, or .

## Linear Pairs

**Adjacent Angles:** Two angles that have the same vertex, share a side, and do not overlap.

and are adjacent.

and are NOT adjacent because they overlap.

**Linear Pair:** Two angles that are adjacent and the non-common sides form a straight line.

and are a linear pair.

**Linear Pair Postulate:** If two angles are a linear pair, then they are supplementary.

**Example 5:** ** Algebra Connection** What is the measure of each angle?

**Solution:** These two angles are a linear pair, so they add up to .

Plug in to get the measure of each angle.

**Example 6:** Are and a linear pair? Are they supplementary?

**Solution:** The two angles are not a linear pair because they do not have the same vertex. They are supplementary, .

## Vertical Angles

**Vertical Angles:** Two non-adjacent angles formed by intersecting lines.

and are vertical angles

and are vertical angles

These angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.

**Investigation 1-6: Vertical Angle Relationships**

- Draw two intersecting lines on your paper. Label the four angles created , and , just like the picture above.
- Use your protractor to find .
- What is the angle relationship between and called? Find .
- What is the angle relationship between and called? Find .
- What is the angle relationship between and called? Find .
- Are any angles congruent? If so, write them down.

From this investigation, you should find that and .

**Vertical Angles Theorem:** If two angles are vertical angles, then they are congruent.

We can prove the Vertical Angles Theorem using the same process we used in the investigation. We will not use any specific values for the angles.

From the picture above:

All of the equations , so Equation 1 = Equation 2 and Equation 2 = Equation 3.

Cancel out the like terms

Recall that anytime the measures of two angles are equal, the angles are also congruent. So, and too.

**Example 7:** Find and .

**Solution:** is vertical angles with , so .

is a linear pair with or , so .

.

**Know What? Revisited** The compass has several vertical angles and all of the smaller angles are . Directions that are opposite each other have the same angle measure, but of course, a different direction. All of the green directions have the same angle measure, , and the purple have the same angle measure, . and all have different measures, even though they are all apart.

## Review Questions

- Questions 1 and 2 are similar to Examples 1, 2, and 3.
- Questions 3-8 are similar to Examples 3, 4, 6 and 7.
- Questions 9-16 use the definitions, postulates and theorems from this section.
- Questions 17-25 are similar to Example 5.

- Find the measure of an angle that is complementary to if is
- Find the measure of an angle that is supplementary to if is

Use the diagram below for exercises 3-7. Note that .

- Name one pair of vertical angles.
- Name one linear pair of angles.
- Name two complementary angles.
- Name two supplementary angles.

- What is:
- If , find:

For 9-16, determine if the statement is true or false.

- Vertical angles are congruent.
- Linear pairs are congruent.
- Complementary angles add up to .
- Supplementary angles add up to
- Adjacent angles share a vertex.
- Adjacent angles overlap.
- Complementary angles are always .
- Vertical angles have the same vertex.

For 17-25, find the value of or .

- Find .
- Find .