I do not understand what the notation "$\operatorname{mod}{2x}$" or "$\operatorname{mod}{x}$" means in a geometric proof involving angles.
I need to understand a proof of a lemma about the Illumination Problem in Euclidian Space; the proof's author is George Tokarsky (1991). See the images below. Can you help me understand?
Thank you, very much! ;)
We are given that $x$ divides $90.$ Usually we only make such statements in the case where $x$ is an integer, and presumably if we were to look at the larger context of the problem where it appears in the book, it would be clear that we are meant to interpret it so that $x$ is an integer.
We are also told about a triangle that has one angle measuring $x$ degrees and one angle measuring $nx$ degrees. Now observe carefully: $x$ degrees is not an angle, it is merely the measure of an angle; and $x$ without the word "degrees" attached to it is not the same thing as $x$ degrees. By itself, $x$ is neither an angle nor even the measure of an angle; $x$ is just a number, specifically, an integer.
So when you see something like $\theta \equiv 90 \mod 2x$ in the proof, it is a lot like when we write something like $y \equiv 11 \mod 30.$ Of course there is only ever one integer named $30,$ whereas $2x$ could be any of several different possible integers, but still $2x$ is a positive integer and can be used as a modulus.
That's what it means. To tell whether all the statements that are made regarding the integer numbers of degrees in various angles are true, you have to consider the geometry of triangles and of pool shots. But that seems to me to be the topic of a different question.