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# Angle Classification

## Categories of angles based on measurements and relationships.

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Angle Classification

### Classifying Angles

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*}.

#### Classifying an Angle

1. 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.

2. 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.

3. 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.

### Examples

Name each type of angle:

#### Example 1

\begin{align*}90^\circ\end{align*}

This angle is exactly \begin{align*}90^\circ\end{align*}, so it is right.

#### Example 2

\begin{align*} 67^\circ\end{align*}

\begin{align*} 67^\circ\end{align*} is less than \begin{align*}90^\circ\end{align*}, so it is acute.

#### Example 3

\begin{align*} 180^\circ\end{align*}

This angle is exactly \begin{align*}180^\circ\end{align*}, so it is straight.

### Review

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

1. Two angles always add up to be greater than \begin{align*}90^\circ\end{align*}.
2. \begin{align*}180^\circ\end{align*} is an obtuse angle.
3. \begin{align*}180^\circ\end{align*} is a straight angle.
4. Two perpendicular lines intersect to form four right angles.
5. The measure of a right angle and an acute angle sum to the measure of an obtuse angle.

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

1. \begin{align*}55^\circ\end{align*}
2. \begin{align*}92^\circ\end{align*}
3. \begin{align*}178^\circ\end{align*}
4. \begin{align*}5^\circ\end{align*}
5. \begin{align*}120^\circ\end{align*}
6. \begin{align*}73^\circ\end{align*}
7. Interpret the picture to the right. Write down all equal angles, segments and if any lines are perpendicular.
8. 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.

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

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### Vocabulary Language: English

Acute Angle

An acute angle is an angle with a measure of less than 90 degrees.

Obtuse angle

An obtuse angle is an angle greater than 90 degrees but less than 180 degrees.

Perpendicular

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle. The product of the slopes of two perpendicular lines is -1.

Right Angle

A right angle is an angle equal to 90 degrees.