# 8.4: Adjacent and Vertical Angles

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

### Let's Think About It

Tania is working for a local fair. She has to help her coworkers assemble miniature replicas of the Ferris wheel. The replicas will be given to every 50th kid who rides on the Ferris wheel. As she is assembling the replicas, she notices a pattern with the angles. She notices that she can predict the measure of the angles that are across from each other. If one of the angles is

In this concept, you will learn about the relationships among angles formed by intersecting lines.

### Guidance

Intersecting lines form angles.

**Adjacent angles** are angles that share the same vertex and one common side. If they combine to make a straight line, their measures must add up to

Angles 1 and 2 form a straight line, so their measurements will add up to

The sum of each angle pair is

This pattern of adjacent angles forms whenever two lines intersect. Notice that the two angles measuring **vertical angles**. Vertical angles are always equal.

Let's look at an example.

Identify all of the pairs of adjacent angles and the two pairs of vertical angles in the figure below.

Angles

Now let’s look at line

Angles

The measure of one angle can be used to find the measure of the second angle.

Find the measure of angle

One angle measures

First, determine how these two angles are related.

The two angles are opposite of each other.

Next, decide if the angles are adjacent or vertical.

The two angles are vertical angles.

Then, use the relationship to determine the value of angle

Angle

The answer is that angle

Here is another example.

Find the measure of

Find how the known angle and the unknown angle are related. This time angle

The answer is that angle

### Guided Practice

Solve the following problem.

Two lines intersect. One vertical angle has a measure of

First, note the relationship between the two angles.

The angles are adjacent and form a straight line.

Next, set up an equation.

Unknown angle +

Then solve for the unknown angle.

Unknown angle

The answer is that the adjacent angle has a measure of

### Examples

#### Example 1

Two adjacent angles form a straight line. If one of the angles is

First, note the relationship between the two angles.

The angles are adjacent.

Next, set up an equation.

Other angle

Then, solve for the missing angle.

Other angle =

The answer is that the other angle measures

#### Example 2

Angle

First, note the relationship between the two angles.

The angles are vertical.

Next, state the relationship between vertical angles.

Vertical angles are equal.

Then, write out the answer.

Angle

The answer is that the measure of angle \begin{align*}C\end{align*}

#### Example 3

Angle \begin{align*}A\end{align*} and angle \begin{align*}D\end{align*} are adjacent angles that form a straight line. If angle \begin{align*}A\end{align*} measures \begin{align*}37^o\end{align*}, what is the measure of angle \begin{align*}D\end{align*}?

First, note the relationship between the two angles.

The angles are adjacent and form a straight line.

Next, set up an equation.

\begin{align*}37^o + D = 180^o\end{align*}

Then solve for the missing angle.

\begin{align*}D = 180^o - 37^o = 143^o\end{align*}

The answer is that angle \begin{align*}D\end{align*} measures \begin{align*}143^o\end{align*}.

### Follow Up

Remember Tania's replica of the Ferris wheel? If one of the angles is \begin{align*}30^o\end{align*} , what is the measure of the vertical angle?

First, note the relationship between the two angles.

The angles are vertical.

Next, state the relationship between vertical angles.

Vertical angles are equal.

Then, write out the answer.

The vertical angle is \begin{align*}30^o\end{align*}.

The answer is that the measure of vertical angle is \begin{align*}30^o\end{align*} .

### Video Review

### Explore More

Identify whether the lines below are parallel, perpendicular, or just intersecting.

1.

2.

3.

4.

5. Lines that will never intersect.

6. Lines that intersect at a \begin{align*}90^\circ\end{align*} angle.

7. Lines that cross at one point.

Tell whether the pairs of angles are adjacent or vertical.

8.

9.

10.

11.

12. Two angles with the same measure.

13. An angle next to another angle.

14. An angle that is congruent to another angle.

15. Two angles with different measures whose sum is \begin{align*}180^\circ\end{align*}.

Find the measure of the unknown angle.

16.

17.

18.

19.

20.

### Image Attributions

In this concept, you will learn about the relationships among angles formed by intersecting lines.

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