Definition:Angle

Definition
Given two intersecting lines or line segments, the amount of rotation about the intersection required to bring one into correspondence with the other is called the angle between them.

Euclid defined this concept as:

"A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."

Note the distinction made here between angles lying in a plane and those that (implicitly) do not.

Subtend
Let $$AB$$ be a line segment and $$C$$ be a point:



The line segment $$AB$$ is said to subtend the angle $$\angle ACB$$.

Rectilineal
As defined by Euclid:

"And when the lines containing the angle are straight, the angle is called rectilineal."

Thus the distinction is made between straight-line angles and curved-line angles.

Most of the time the fact that angles are rectilineal is taken for granted.

Measurement
Angles are usually measured in degrees, denoted by $$^\circ$$, or in radians, denoted by rad or without any unit. One full rotation is $$360^\circ$$ and $$2\pi$$ rad.

Types of Angle
Angles can be divided into categories:
 * Zero angle
 * Acute angles
 * Right angles
 * Obtuse angles
 * Straight angles
 * Reflex angles
 * Full angles

It is possible to have angles outside the $$[0^\circ\,. \, . \,360^\circ ]$$ or $$[0\,. \, . \,2\pi ]$$ range, but in geometric contexts it is usually preferable to convert these to angles inside this range by adding or subtracting multiples of $$360^\circ$$ or $$2\pi$$.

Directed versus Undirected Angles
The most basic definition of angle is an undirected angle on the interval $$[0^\circ\,. \, . \,180^\circ]$$ or $$[0\,. \, . \,\pi]$$. This definition is often insufficient, in cases such as the external angles of a polygon. Therefore, angles are most commonly defined in one of two ways:
 * Undirected angles on the interval $$[0^\circ \, . \, . \,360^\circ ]$$ or $$[0\, . \, . \,2\pi ]$$.
 * Directed angles, with the positive direction being counterclockwise from a given line (or, if no line is specified, from the x-axis). This definition is more commonly found in applied mathematics, such as in surveying, navigation, or, more colloquially, in a $$720^\circ$$ degree spin in skateboarding, skiing, etc.

Note
It is advisable to use radians, especially in calculus, since the derivatives of trigonometric functions work out without the necessity of multiplicative constants when angles are measured in radians (such as $$\frac{d}{dx}\sin x=\cos x$$).