Angle

When two straight lines come together, they make an angle. The two lines are called the sides^[1] of the angle, and they meet at a point. A flat surface (called a plane) also forms an angle when it meets another.

To represent an angle, Greek letters such as $\alpha$ (alpha), $\beta$ (beta), $\gamma$ (gamma) and $\theta$ (theta) are sometimes used.^[2] An angle indicates the space between its sides, or the amount of rotation needed to make one side coincide the other.^[3]^[4]

To measure the size of an angle, we use units called degrees. A degree is a standard unit and we use the symbol ° after a number to show that it is a number of degrees. We can use a decimal number or a fraction for part of a degree, but a degree can also be divided into 60 minutes (1° = 60'), and a minute can be divided into 60 seconds (1' = 60"). So 22.5°, 221⁄2° and 22° 30' are all the same angle.

In mathematics, angles can also be (and often are) measured in radians instead of degrees, by using the conversion factor $2\pi {\mbox{ rad}}=360^{\circ }$ (for example, $22.5^{\circ }={\tfrac {\pi }{8}}{\mbox{ rad}}$ ). Yet another unit of angle is gradian,^[4] with $100{\text{ grad}}=90^{\circ }$ .

Angles are studied in geometry, where an angle where edges meet is often called a vertex. For example, the three sides of a triangle are its edges and two of the edges meet at each vertex. Similarly, two of the six sides (or faces) of a cube meet at each of its twelve edges, and three edges meet at each of its eight corners (or vertices, which is the plural version of vertex).

Individual angles

In a zero angle the lines lie one upon the other thus creating a 0° angle aka the zero angle.

An angle measuring greater (wider) than 0° but less (narrower) than 90° is called an acute angle.

An angle 90° wide is called a right angle.

An angle more than 90° but less than 180° is called an obtuse angle.

An angle that measures 180° is called a straight angle.
An angle wider than 180° and narrower than 360° is called a reflex angle.
An angle that has a made or full circle/completed 360° is called a full or complete angle.^[5]

Special angle pairs

In geometry, there are pairs angles having a special relationship with each other, making them interesting and convenient.

There is a pair of angles called complementary angles to which the sum of their measure (wideness) is equal to one right angle (which is equal to ${\tfrac {1}{4}}$ turn, 90°, or ${\tfrac {\pi }{2}}$ radians). Supplementary angles are also two angles, this time their combined measure is a straight angle ( ${\tfrac {1}{2}}$ turn, 180°, or $\pi$ radians). Two angles that total to a full angle ( $1$ turn, 360°, or $2\pi$ radians) are referred to as explementary or conjugate angles.