Do you know about adjacent and vertical angles? Take a look at this illustration.

If you were asked to name all of the vertical angles and adjacent angles would you know what to identify? This Concept will teach you how to identify these special angle relationships.

### Guidance

When lines intersect, they create special relationships between the angles that they form. Once we understand these relationships, we can use them to find the measure of angles formed by the intersecting lines.

*Adjacent angles***are angles that share the same vertex and one common side. If they combine to make a straight line, adjacent angles must add up to \begin{align*}180^{\circ}\end{align*}.** The word “adjacent” means “next to” that can help you to remember adjacent angles.

Below, angles 1 and 2 are adjacent. Angles 3 and 4 are also adjacent.

Can you see that angles 1 and 2, whatever their measurements are, 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. **Because the adjacent angle pairs form lines, we can also say that they are supplementary.** They must add up to \begin{align*}180^{\circ}\end{align*}.

We can find the sums in this way.

\begin{align*}\angle{1} + \angle{2} = 180^{\circ}\\ \angle{3} + \angle{4} = 180^{\circ}\\\end{align*}

**As you work through this lesson, you will find that some information leads you to other information. Here is the first example of that. Whenever adjacent angles form a straight line, they are also supplementary. The sum of their angles will be \begin{align*}180^{\circ}\end{align*}.**

Notice that when there are two angles next to each other, there are also two angles diagonally across from each other. These are called *vertical angles.**Vertical angles***are angles that are diagonally across from each other and have the same measure.**

These relationships always exist whenever any two lines intersect. Look carefully at the figures below. Understanding the four angles formed by intersecting lines is a very important concept in geometry.

**Identify the vertical angles and the adjacent angles in the diagram below.**

**First, think back to the definition of adjacent and vertical angles.**

**Adjacent angles are next to each other. When they form a straight line they are supplementary angles. We can see from the diagram that angles 1 and 3 are adjacent. Angles 2 and 4 are also adjacent.**

**Vertical angles are diagonal from each other and have the same measure. In this case, angles 1 and 4 are vertical. Angles 2 and 3 are also vertical angles.**

#### Example A

True or false. Adjacent angles are across from each other.

**Solution: False. Adjacent angles are next to each other.**

#### Example B

True or false. Pairs of vertical angles have the same measure.

**Solution: True.**

#### Example C

True or false. Adjacent angles can also be vertical angles.

**Solution: False.**

Now let's go back to the dilemma from the beginning of the Concept. Here is the illustration once again.

First identify the vertical angles.

**Angles 1 and 3 are vertical angles. Angles 2 and 4 are also vertical angles.**

Next, identify the adjacent angles. Remember that adjacent angles have a sum of 180^{\circ}.

**Angles 1 and 2, angles 1 and 4, angles 4 and 3, and angles 2 and 3 are all adjacent angles.**

### Vocabulary

- Intersecting lines
- lines that cross at one point.

- Adjacent Angles
- angles that are next to each other.

- Vertical Angles
- angles that are diagonally across from each other.

- Angle
- the measure of the space formed by two intersecting lines.

### Guided Practice

Here is one for you to try on your own.

Find \begin{align*}m\angle 1\end{align*}.

**Solution**

First, let's think about what we know about the angles identified.

\begin{align*}\angle 1\end{align*} is vertical angles with \begin{align*}18^\circ\end{align*}, so \begin{align*}m\angle 1 = 18^\circ\end{align*}.

**This is our answer.**

### Video Review

Khan Academy Proving Vertical Angles are Equal

### Practice

Directions: Identify whether each angle pair can be classified as adjacent angles or vertical angles or neither.

1.\begin{align*}\angle\end{align*}INK and \begin{align*}\angle\end{align*}MNL

2. \begin{align*}\angle\end{align*}INJ and \begin{align*}\angle\end{align*}NJK

3. \begin{align*}\angle\end{align*}MNL and \begin{align*}\angle\end{align*}LNK

4. \begin{align*}\angle\end{align*}JNL and \begin{align*}\angle\end{align*}INM

5. \begin{align*}\angle\end{align*}INM and \begin{align*}\angle\end{align*}KNL

6. If \begin{align*}m\angle INJ = 63^\circ\end{align*}, find \begin{align*}m\angle MNL\end{align*}.

Directions: Use this diagram to answer the following questions.

7. True or false. \begin{align*}\angle1\end{align*} and \begin{align*}\angle2\end{align*} are adjacent angles.

8. What is the measure of \begin{align*}\angle1\end{align*}?

9. What is the measure of \begin{align*}\angle2\end{align*}?

10. What is the relationship between \begin{align*}\angle2\end{align*} and the angle opposite it?

11. True or false. Adjacent angles 1 and 2 form a straight line with a value of \begin{align*}180^{\circ}\end{align*}.

Directions: Answer true or false for each question.

12. Supplementary angles are also vertical angles.

13. Vertical angles have the same measure.

14. Adjacent angles always have a sum of \begin{align*}180^{\circ}\end{align*}.

15. Adjacent angles are also vertical angles.

16. Adjacent angles are formed when lines intersect.