An angle is defined as two rays $\overrightarrow{YX}$, $\overrightarrow{YZ}$ sharing a midpoint $Y$; the angle formed is called $\angle XYZ$ or simply $\angle Y$. Then, the measure of $\angle Y$ is denoted ${\small m}\angle Y$.
We know the various definitions concerning the classification of angles. What interests me is the special case of ${\small m}\angle Y=0^\circ$. I asked my teacher what this would form. In response, he told me it would form two overlapping rays. Since these two rays are equal, he said that the instance was trivial, uninteresting, and not practical in any proofs or the like.
My question is thus: Would an angle with a measure of $0^\circ$ be of any practical use? The only use I could think of is this:
O==========O======O===>
Y X Z
Where the distinction of an extra point is required. However, this seems trivial to me, and it probably is. If there is some proof it is used in, please provide a reference.
Edit
Since some nice person dropped a downvote without a comment, it is mere guesswork as to the problem. Ergo, I will do my best.
This is merely a recreational question. I have a decent exposure in trig (what I don't know, I can learn) and a deal of knowledge in the art of proofs, logic, set theory, number theory, and (of course) geometry.
I am aware of trigonomerric identities, relavently the facts that $\sin\left(0^\circ\right)=\sin\left(360^\circ\right)$; is this relavent?
Thank you.
I assume you haven't been exposed to much trigonometry yet. There you will see the concept of 0 angles, and negative angles, and angles with sizes above $360^\circ$.
From a strictly geometric viewpoint, these are not necessary. But from an analytic viewpoint, they are essential. For example, consider 2 planets moving in a perfectly circular orbits about their sun. There are a number of reasons it would be useful to know the angle between them. But there are times when that angle is $0^\circ$. So there already, we need a $0$ angle, and its meaning is not trivial. Indeed, its meaning is a very important condition. And $360^\circ$, which is the exact same angle, is obviously not trivial, otherwise it wouldn't be so familiar to you. We mention it regularly exactly because it is a useful concept.
If we examine how the angle between planets changes with time, having it go back to $0$ every time it reaches $360$ is troublesome. It means instead of nicely behaved functions, we have what are called "jump discontinuities", which are a mess to deal with. This is why it is useful to expand the definition of angle to extend beyond $0$ and $360$. Then everything continues to behave nicely.