What if you wanted to group different angles into different categories? After completing this Concept, you'll be able to classify angles geometrically.

### Watch This

CK-12 Foundation: Chapter1AngleClassificationA

James Sousa: Animation of Types of Angles

### Guidance

By looking at the protractor we measure angles from \begin{align*}0^\circ\end{align*} to @$\begin{align*}180^\circ\end{align*}@$. Angles can be classified, or grouped, into four different categories.

**Straight Angle:** When an angle measures @$\begin{align*}180^\circ\end{align*}@$. The angle measure of a straight line. The rays that form this angle are called opposite rays.

**Right Angle:** When an angle measures @$\begin{align*}90^\circ\end{align*}@$.

Notice the half-square, marking the angle. This marking is always used to mark right, or @$\begin{align*}90^\circ\end{align*}@$, angles.

**Acute Angles:** Angles that measure between @$\begin{align*}0^\circ\end{align*}@$ and @$\begin{align*}90^\circ\end{align*}@$.

**Obtuse Angles:** Angles that measure between @$\begin{align*}90^\circ\end{align*}@$ and @$\begin{align*}180^\circ\end{align*}@$.

It is important to note that @$\begin{align*}90^\circ\end{align*}@$ is NOT an acute angle and @$\begin{align*}180^\circ\end{align*}@$ is NOT an obtuse angle.

Any two lines or line segments can intersect to form four angles. If the two lines intersect to form right angles, we say the lines are **perpendicular**.

The symbol for perpendicular is @$\begin{align*}\bot\end{align*}@$, so these two lines would be labeled @$\begin{align*}l \bot m\end{align*}@$ or @$\begin{align*}\overleftrightarrow{A C} \bot \overleftrightarrow{D E}\end{align*}@$.

There are several other ways to label these two intersecting lines. This picture shows **two perpendicular lines, four right angles, four** @$\begin{align*}90^\circ\end{align*}@$ **angles**, and even **two straight angles,** @$\begin{align*}\angle ABC\end{align*}@$ and @$\begin{align*}\angle DBE\end{align*}@$.

#### Example A

Name the angle and determine what type of angle it is.

The vertex is @$\begin{align*}U\end{align*}@$. So, the angle can be @$\begin{align*}\angle TUV\end{align*}@$ or @$\begin{align*}\angle VUT\end{align*}@$. To determine what type of angle it is, compare it to a right angle. Because it opens wider than a right angle and less than a straight angle it is **obtuse**.

#### Example B

What type of angle is @$\begin{align*}165^\circ\end{align*}@$?

@$\begin{align*}165^\circ\end{align*}@$ is greater than @$\begin{align*}90^\circ\end{align*}@$, but less than @$\begin{align*}180^\circ\end{align*}@$, so it is **obtuse**.

#### Example C

What type of angle is @$\begin{align*}84^\circ\end{align*}@$?

@$\begin{align*}84^\circ\end{align*}@$ is less than @$\begin{align*}90^\circ\end{align*}@$, so it is **acute**.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter1AngleClassificationB

### Guided Practice

Name each type of angle:

1. @$\begin{align*}90^\circ\end{align*}@$

2. @$\begin{align*} 67^\circ\end{align*}@$

3. @$\begin{align*} 180^\circ\end{align*}@$

**Answers:**

1. Right

2. Acute

3. Straight

### Interactive Practice

### Explore More

For exercises 1-5, determine if the statement is true or false.

- Two angles always add up to be greater than @$\begin{align*}90^\circ\end{align*}@$.
- @$\begin{align*}180^\circ\end{align*}@$ is an obtuse angle.
- @$\begin{align*}180^\circ\end{align*}@$ is a straight angle.
- Two perpendicular lines intersect to form four right angles.
- A right angle and an acute angle make an obtuse angle.

For exercises 6-11, state what type of angle it is.

- @$\begin{align*}55^\circ\end{align*}@$
- @$\begin{align*}92^\circ\end{align*}@$
- @$\begin{align*}178^\circ\end{align*}@$
- @$\begin{align*}5^\circ\end{align*}@$
- @$\begin{align*}120^\circ\end{align*}@$
- @$\begin{align*}73^\circ\end{align*}@$
- Interpret the picture to the right. Write down all equal angles, segments and if any lines are perpendicular.
- Draw a picture with the following requirements.

@$$\begin{align*}& AB = BC = BD && m \angle ABD = 90^\circ\\ & m \angle ABC = m \angle CBD && A, B, C \ \text{and} \ D \ \text{are coplanar} \end{align*}@$$

In 14 and 15, plot and sketch @$\begin{align*}\angle ABC\end{align*}@$. Classify the angle. Write the coordinates of a point that lies in the interior of the angle.

- @$\begin{align*}A(5, -3), B(-3, -1), C(2, 2)\end{align*}@$
- @$\begin{align*}A(-3, 0), B(1, 3), C(5, 0)\end{align*}@$