# Describing Angles

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## Angle

When two rays share an endpoint, an **angle** is formed. Below, the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ create angle $ABC.$

When referring to a specific angle, the notation $\angle$ is often used to name the angle. Additionally, $m\angle$ is used to note the size or measure of an angle, which can be interpreted as the amount of turn in the rotation between the rays. It is often measured in degrees, but can also be measured in radians. An angle of $0 ^\circ$ shows no rotation, while an angle of $360 ^\circ$ shows a complete rotation.

Sometimes, the space outside an angle — rather than inside — is of interest. In figure above, $\alpha$ measures the*interior angle*formed by the rays, whereas $\beta$ measures the

*exterior angle.*

## Measuring an Angle with a Protractor

A protractor is a tool that can be used to measure angles in degrees.

The measure of the angle below will be found using a protractor.

To determine the measure of an angle, one of its rays — it doesn't matter which — is placed along the bottom of the protractor so that it is aligned with $0.$ The intersection of the rays is placed in the middle, as shown.

The ray intersects the zero on the right side of the protractor, which means the inner orientation, which increases in the counterclockwise direction, is used to determine the angle.

The other ray is located at $145,$ according to the inner orientation. Therefore, the angle measures $145^\circ.$## Angle Bisector

An angle bisector is a ray that divides an angle into two angles that have the same measure.

## Congruent Angles

Angles that have the same measure are said to be congruent. In a figure, congruent angles are usually indicated by the same number of arcs. In the figure below, $\angle A$ and $\angle C$ are congruent, which can be written as $\angle A \cong \angle C.$

## Types of Angles

## Pairs of Angles

Pairs of angles can be classified in different ways depending on how their measures relate.

### Complementary Angles

Two angles whose measures add to $90 ^\circ$ are called complementary angles.

In the figure above, $\angle CBD$ is complementary to $\angle DBA$ because

$m\angle CBD + m\angle DBA = 90 ^\circ.$### Supplementary Angles

*linear pair*, and together they form a straight angle.

In the figure above, $\angle \alpha$ and $\angle \beta$ are supplementary because

$m\angle \alpha + m\angle \beta = 180 ^\circ.$### Vertical Angles

When two lines or line segments intersect, vertical angles are formed on opposite sides of the point of intersection. In the figure below, vertical angles are marked with the same number of hatch marks.

## Vertical Angles Theorem

The Vertical Angles Theorem states that $\text{vertical angles are congruent.}$ In the figure below, this means that $\angle 1 \cong \angle 3$ and $\angle 2 \cong \angle 4.$

This can be proven using supplementary angles.## Exercises

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