Best way to describe the angle from another ray on a circle

Question

What I have is $3$ rays each of whose origin is the same.

Ray $Z$ is always $0^{\circ}$ and ray $X$ and ray $Y$ is a random angle away from ray $Z$ in a clockwise direction.

What is the best way to get the shortest Angle between ray $X$ and Ray $Y$.

Answer 1

You can always rotate the circle so that ray $z$ lies on top of where you would consider $0^\circ$ to normally be. This rotation wouldn't affect the angles. Then you just get the angle between $x$ and $y$ by subtracting the difference as usual.

A simple proof of this is as follows. Let $x_z$ be the angle of $x$ in terms of $z$ and let $y_z$ be the angle of $y$ in terms of $z$. Let's also use $x,y,z$ to denote the angles of the respective rays according where you consider $0^\circ$ to be. We want to find $x-y$. $$x-y=(x-z)-(y-z)=x_z-y_z$$