Angle

The names, intervals, and measuring units are shown in the table below:

When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.

If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle.

The angle addition postulate states that if B is in the interior of angle AOC, then

The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

Other units used to represent angles are listed in the following table. These units are defined such that the number of turns is equivalent to a full rotation.

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

For an angular unit, it is definitional that the angle addition postulate holds. Some angle measurements where the angle addition postulate does not hold include:

These measurements clearly depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with