# What Are Adjacent Angles? - Definition & Examples

## Adjacent angles definition

Adjacent angles are a pair of angles that share a common side and vertex.

Three features make adjacent angles easy to pick out:

Adjacent angles exist as pairs

They share a common vertex

They share a common side

If the two angles only share a common vertex, then they are vertical angles. Vertical angles are a pair of opposite angles made by two intersecting lines.

### What is a common vertex?

A **common vertex** is a vertex that is shared by two angles. A vertex is the point at the intersection of any two linear constructions.

Line

Line segment

Ray

You can mix and match these to create **vertices** (the plural of vertex) in many ways:

You see vertices in the corners of polygons, as central angles in circles, and when linear constructions, like parallel lines and transversals, cross.

### What is a common side in geometry?

A **common side** is one line, ray, or line segment used to create two angles sharing the same vertex. Both angles use the common side and one other side.

Adjacent angles are always pairs and never overlap.

Let's see how one vertex of a square can demonstrate adjacent angles.

Here we have a simple square formed by four sides creating four vertices: **∠W**, **∠H**, **∠I**, and **∠Z**.

If we connect **Point W** with **Point I**, we construct **diagonal WI**. This creates two additional angles at **Point W**:

**∠ZWI****∠HWI**

Notice both those angles share a common vertex at **Point W**, and a common side, line segment **WI**. Angles **∠ZWI** and **∠HWI** are adjacent angles.

## Linear pairs

When a pair of adjacent angles create a straight line or straight angle, they are a linear pair. The sum of their angles is **180°** or **π** radians.

Angles that sum to **180°** are called supplementary angles. Here is a linear pair. See if you can identify the common side and common vertex:

**Ray AT** is the common ray of both angles. Did you identify **∠A** as the common vertex?

## Parallel lines and transversals

Here are parallel lines **CP** and **MN** cut by transversal **IK**. Where the transversal cuts across them, we have points **H** and **U**:

Not only does this construction form eight pairs of angles (adjacent angles), but all those pairs are also linear pairs! Which angles are adjacent angles?

**∠CHI**and**∠PHI****∠CHI**and**∠CHU****∠PHI**and**∠PHU****∠CHU**and**∠PHU****∠MUK**and**∠NUK****∠MUH**and**∠NUH****∠MUH**and**∠MUK****∠NUH**and**∠NUK**

These are all examples of adjacent angles. Adjacent angles can help prove that lines are parallel.

## Adjacent angles example

May 12 is the birthday of Maryam Mirzakhani, a famous mathematician who studied a special kind of geometry called hyperbolic geometry. To celebrate her work, your math club bakes a birthday cake and puts you in charge of slicing it into eighths:

Are all the angles of Maryam's cake adjacent angles?

Well, no. **∠IMY** is adjacent to both **∠RMI** and **∠YMN**, but notice that **∠RMI** is not adjacent to **∠YMN**, even if both angles share **vertex M**.

Angle relationships like adjacent angles must share both a common vertex (**Point M**) and a common side. **∠RMI** shares no common side with **∠YMN**.

Can you find any linear pairs in Maryam's cake? We hope so! For each diameter of Maryam's cake, *three* linear pairs exist!

To see that, we can take just one line segment, **YA**, as an example. You can create the straight line **YA** with these three linear pairs:

**∠YMI**and**∠IMA****∠YMR**and**∠RMA****∠YMZ**and**∠ZMA**

Adjacent angles are two angles sharing a common vertex and a common side. They appear in many places but are prominent in parallel lines cut by transversals.