In short, the integers cover the complete turns around the circle. If the point covered by zero is called the starting point on the circle, the integers count the number of revolutions around the circle from the starting point. The revolutions provide a method of measuring angles. Suppose here that the initial side of the angle is the half line from the center C of the circle. On the circle measured in the counterclockwise direction there is a point that is 1/4 of a single revolution. The terminal side of that angle contains the segment between the center C and that point just located. The measure of the angle then is + 1/4 of a revolution,.Similarly, there are angles 1/2 rev (straight angle), 3/4 rev, etc. The angle - 1/4 rev is measured in the clockwise direction and its terminal side passes through the same point as the terminal side of the angle 3/4 rev.
[x.1] If the pair of sides of two angles measured in the same direction are congruent then the angles are said to be equivalent.
[x.2] Two angles are equivalent if and only if the difference of their angles measured in revolutions is an integer.
The parts of the circle may be divided into any number of parts. A common number is 360. Then the angle with initial side through zero and the terminal side through the first point is by definition 1° (one degree). So 1 rev = 360°, 2 rev = 720° -1 rev = -360° -2 rev = -720°. in general for any integer n, n rev = n x 360°. There is a similar formula for fractional parts of a revolution. 1/4 rev = 360°/4 = 90°, 1/2 rev = 180° -1/12 rev = - 30°
[x.3] Two angles are equivalent if and only if the difference of their angles measured in degrees is a multiple of 360°.
For example, angles measuring 60° 420° 3660° and -300° are all equivalent. Also 1/6 rev is equivalent to all of them.