Suppose a square table. You are shooting from a corner. A ball ends its path upon landing in a corner. At what angles can you shoot a ball from your corner such that the ball will never come back to your corner?
Edit: seemingly my question would be how can you prove that if the trajectory is periodic, how can you prove that it will encounter another "corner" point before it returns to itself?
Imagine replicating your square billiard table over the whole plane. Then "bounces" and "crossing into the next billiard table" become more or less equivalent (there's some flipping-over to consider).
So this question becomes "which rays from the origin contain no point of the integer grid?"
You should be able to turn that into a statment about angles, I believe.