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# Angle Classification

## Categories of angles based on measurements and relationships.

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Angle Classification Student Textbook

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

### 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.

#### Example B

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

#### Example C

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

Watch this video for help with the Examples above.

### Vocabulary

A straight angle is when an angle measures \begin{align*}180^\circ\end{align*}. A right angle is when an angle measures \begin{align*}90^\circ\end{align*}. Acute angles are angles that measure between \begin{align*}0^\circ\end{align*} and \begin{align*}90^\circ\end{align*}. Obtuse angles are angles that measure between \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*}. If two lines intersect to form right angles, the lines are perpendicular.

### Practice

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. A right angle and an acute angle make 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(1, 3), C(5, 0)\end{align*}

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