# 8.4: Adjacent and Vertical Angles

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

**Practice**Vertical Angles

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

### Adjacent and Vertical Angles

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 \begin{align*}180^\circ\end{align*}. Below, angles 1 and 2 are adjacent. Angles 3 and 4 are also adjacent. Adjacent angles can also be thought of as “next to” each other.

Angles 1 and 2 form a straight line, so their measurements will add up to \begin{align*}180^\circ\end{align*}. This is true for angles 3 and 4, because they also form a line. But that’s not all. Angles 1 and 4 also form a line. So do angles 2 and 3. These are also pairs of adjacent angles. Let’s see how this works with angle measurements.

The sum of each angle pair is \begin{align*}180^\circ\end{align*}.

This pattern of adjacent angles forms whenever two lines intersect. Notice that the two angles measuring \begin{align*}110^\circ\end{align*} are diagonal from each other, and the two angles measuring \begin{align*}70^\circ\end{align*} are diagonal from each other. This is the other special relationship among pairs of angles formed by intersecting lines. These angle pairs are called **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 \begin{align*}Q\end{align*} and \begin{align*}R\end{align*} are next to each other and together make a straight angle along line \begin{align*}n\end{align*}. What about \begin{align*}T\end{align*} and \begin{align*}S\end{align*}? They also sit together along line \begin{align*}n\end{align*}. Both are adjacent pairs.

Now let’s look at line \begin{align*}m\end{align*}. Angles \begin{align*}Q\end{align*} and \begin{align*}T\end{align*} make a straight angle along line \begin{align*}m\end{align*}, and so do angles \begin{align*}S\end{align*} and \begin{align*}R\end{align*}. All four of these pairs are adjacent.

Angles \begin{align*}Q\end{align*} and \begin{align*}S\end{align*} are opposite of each other so they are vertical angles. Angles \begin{align*}T\end{align*} and \begin{align*}R\end{align*}, the small angles, are also opposite each other. Therefore they are the other pair of vertical angles.

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

Find the measure of angle \begin{align*}B\end{align*} below.

One angle measures \begin{align*}50^o\end{align*}, now let's find the measure of angle \begin{align*}B\end{align*}.

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 \begin{align*}B\end{align*}.

Angle \begin{align*}B\end{align*} is \begin{align*}50^o\end{align*}.

The answer is that angle \begin{align*}B = 50^o\end{align*}.

Here is another example.

Find the measure of \begin{align*}\angle Q\end{align*} below.

Find how the known angle and the unknown angle are related. This time angle \begin{align*}Q\end{align*} is not opposite the known angle. It is adjacent, because together they form a straight line. Adjacent angles add up to \begin{align*}180^\circ\end{align*}. Let's use the measure of the known angle to solve for angle \begin{align*}Q\end{align*}.

\begin{align*}138^\circ + \angle Q &= 180\\ \angle Q &= 180 - 138\\ \angle Q &= 42^\circ\end{align*}

The answer is that angle \begin{align*}Q\end{align*} must be \begin{align*}42^\circ\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about 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 the vertical angle is \begin{align*}30^o\end{align*}.

#### Example 2

Solve the following problem.

Two lines intersect. One vertical angle has a measure of \begin{align*}45^\circ\end{align*} . What is the measure of the angle adjacent to it?

First, note the relationship between the two angles.

The angles are adjacent and form a straight line.

Next, set up an equation.

Unknown angle + \begin{align*}45^o = 180^o\end{align*}

Then solve for the unknown angle.

Unknown angle \begin{align*}= 180^o - 45^o = 135\end{align*}

The answer is that the adjacent angle has a measure of **\begin{align*}135^o\end{align*}.**

#### Example 3

Two adjacent angles form a straight line. If one of the angles is \begin{align*}105^o\end{align*}, what is the measure of the other angle?

First, note the relationship between the two angles.

The angles are adjacent.

Next, set up an equation.

Other angle \begin{align*}+ 105^o = 180^o\end{align*}

Then, solve for the missing angle.

Other angle = \begin{align*}180^o - 105^o = 75^o\end{align*}

The answer is that the other angle measures \begin{align*}75^o\end{align*}.

#### Example 4

Angle \begin{align*}B\end{align*} and angle \begin{align*}C\end{align*} are vertical angles. If angle \begin{align*}B\end{align*} is \begin{align*}50^o\end{align*}, what is the measure of angle \begin{align*}C\end{align*}?

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 \begin{align*}C = 50^o\end{align*}

The answer is that the measure of angle \begin{align*}C\end{align*} is \begin{align*}50^o\end{align*}.

#### Example 5

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*}.

### Review

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

- Lines that will never intersect.
- Lines that intersect at a \begin{align*}90^\circ\end{align*} angle.
- Lines that cross at one point.

Tell whether the pairs of angles are adjacent or vertical.

- Two angles with the same measure.
- An angle next to another angle.
- An angle that is congruent to another angle.
- Two angles with different measures whose sum is \begin{align*}180^\circ\end{align*}.

Find the measure of the unknown angle.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.4.

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

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