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# Angle Classification

## Categories of angles based on measurements and relationships.

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Angle Classification

What if you were given the degree measure of an angle? How would you describe that angle based on its size? After completing this Concept, you'll be able to classify an angle as acute, right, obtuse, or straight.

### Watch This

CK-12 Classifying an Angle

James Sousa: Animation of Types of Angles

### Guidance

Angles can be grouped into four different categories.

Straight Angle: An angle that measures exactly \begin{align*}180^\circ\end{align*}.

Acute Angles: Angles that measure between \begin{align*}0^\circ\end{align*} and up to but not including \begin{align*}90^\circ\end{align*}.

Obtuse Angles: Angles that measure more than \begin{align*}90^\circ\end{align*} but less than \begin{align*}180^\circ\end{align*}.

Right Angle: An angle that measures exactly \begin{align*}90^\circ\end{align*}.

This half-square marks right, or \begin{align*}90^\circ\end{align*}, angles. When two lines intersect to form four right angles, the lines are perpendicular. The symbol for perpendicular is \begin{align*}\perp\end{align*}.

Even though all four angles are \begin{align*}90^\circ\end{align*}, only one needs to be marked with the half-square. \begin{align*}l \perp m\end{align*} is read line \begin{align*}l\end{align*} is perpendicular to line \begin{align*}m\end{align*}.

#### Example A

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.

#### Example B

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 is less than a straight angle, it is obtuse.

#### Example C

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.

CK-12 Classifying an Angle

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### 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

### Explore More

For exercises 1-4, 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.

For exercises 5-10, 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*}

In exercises 11-15, use the following information: \begin{align*}Q\end{align*} is in the interior of \begin{align*}\angle ROS\end{align*}. \begin{align*}S\end{align*} is in the interior of \begin{align*}\angle QOP\end{align*}. \begin{align*}P\end{align*} is in the interior of \begin{align*}\angle SOT\end{align*}. \begin{align*}S\end{align*} is in the interior of \begin{align*}\angle ROT\end{align*} and \begin{align*}m\angle ROT = 160^\circ, \ m\angle SOT = 100^\circ\end{align*}, and \begin{align*}m\angle ROQ = m\angle QOS = m\angle POT\end{align*}.

1. Make a sketch.
2. Find \begin{align*}m\angle QOP\end{align*}.
3. Find \begin{align*}m\angle QOT\end{align*}.
4. Find \begin{align*}m\angle ROQ\end{align*}.
5. Find \begin{align*}m\angle SOP\end{align*}.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.6.

### Vocabulary Language: English Spanish

Acute Angle

Acute Angle

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

Obtuse angle

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

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

Right Angle

A right angle is an angle equal to 90 degrees.
Straight angle

Straight angle

A straight angle is a straight line equal to $180^{\circ}$.
vertical angles

vertical angles

Two non-adjacent angles formed by intersecting lines.
Riemann sum

Riemann sum

A Riemann sum is an approximation of the area under a curve, calculated by dividing the region up into shapes that approximate the space.

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