For the “special” right triangles you learn in high school, like 1-1-sqrt(2) and 1-2-sqrt(3), the subtended angles are all rational multiples of pi: pi/3, pi/4, and pi/6. But these triangles are irrational.
What about rational right triangles, like 3-4-5? The angle T adjacent to 3 thus satisfies cos(T) = 3/5. Is T a rational multiple of pi? What about angles in other rational right triangles?