Angles

The measure of rotation between two rays sharing a common endpoint — the foundation of shape and direction.

Practice exercises →

Angle types are classified by their measure

Acute

less than 90°

Right

exactly 90°

Obtuse

between 90° and 180°

Straight

exactly 180°

Vertical

opposite angles match

Definition

An angle is formed when two rays share the same starting point, called the vertex. The rays are the sides of the angle. An angle measures the amount of rotation from one ray to the other.

Angles are measured in degrees. One full rotation is $360°$ .

Common angle types:

Acute angle: between $0°$ and $90°$
Right angle: exactly $90°$ (the little square symbol marks these)
Obtuse angle: between $90°$ and $180°$
Straight angle: exactly $180°$ — a flat line

Definition

Special angle pairs:

Complementary angles add up to $90°$
Supplementary angles add up to $180°$
Vertical angles are the opposite angles formed when two lines cross — they are always equal

Finding a missing angle

Angles $A$ and $B$ are supplementary. If $A = 65°$ , find $B$ .

Since $A + B = 180°$ : $B = 180° - 65° = 115°$

Try it

Two angles are complementary. One angle is three times the other. Find both angles.

Solution

Let the smaller angle be $x$ . Then the larger is $3x$ .

$x + 3x = 90°$ , so $4x = 90°$ , giving $x = 22.5°$ and $3x = 67.5°$ .

Related concepts

Needs first

Points, Lines & Planes

Triangles

Uses this

Geometric Proofs

View in full concept graph →