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.

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.

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, $\angle W$, $\angle H$, $\angle I$, and $\angle Z$.

If we connect $PointW$ with $PointI$, we construct diagonal $WI$. This creates two additional angles at $PointW$:

- $\angle ZWI$
- $\angle HWI$

Notice both those angles share a common vertex at $PointW$, and a common side, line segment $WI$. Angles $\angle ZWI$ and $\angle HWI$ are adjacent angles.

When a pair of adjacent angles create a straight line or straight angle, they are a linear pair. The sum of their angles is $180\xb0$ or $\pi $ radians.

Angles that sum to $180\xb0$ are called supplementary angles.

Here is a linear pair. See if you can identify the common side and common vertex:

$RayAT$ is the common ray of both angles. Did you identify $\angle A$ as the common vertex?

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?

- $\angle CHI$ and $\angle PHI$
- $\angle CHI$ and $\angle CHU$
- $\angle PHI$ and $\angle PHU$
- $\angle CHU$ and $\angle PHU$
- $\angle MUK$ and $\angle NUK$
- $\angle MUH$ and $\angle NUH$
- $\angle MUH$ and $\angle MUK$
- $\angle NUH$ and $\angle NUK$

These are all examples of adjacent angles.

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. $\angle IMY$ is adjacent to both $\angle RMI$ and $\angle YMN$, but notice that $\angle RMI$ is not adjacent to $\angle YMN$, even if both angles share vertex $M$.

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

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:

- $\angle YMI$ and $\angle IMA$
- $\angle YMR$ and $\angle RMA$
- $\angle YMZ$ and $\angle ZMA$

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.

Learn more about the different types of angles, such as interior angles, exterior angles, and complementary angles.

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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