It seems that it could be possible for the dihedral angles of the platonic solids to be rational if one were to stop using our biased degree units and use the units natural to the platonic realm: radians.
Obviously, the cube is 1/4pi radians already and serves as one example, but will the other solids show up with rational multiples of pi?
Also, to support my cause, there is a rule for the angles of each vertex in a closed n-gon (pi*(1-(2/n))) for 2 dimensions -- which always gives rational factors for any number of sides. There must be something similar for 3 dimensions, yes? OR why would it jump from rational numbers in 2d to irrational numbers in 3d?
I assume that by a rational angle in the unit of radians you still mean a rational multiple of $\pi$, as otherwise already the cube is a counterexample. If so, then the answer still is: unfortunately not.
Check out this list of dihedral angles in platonic solids. The angle of the tetrahedron is $\arccos(1/3)$, i.e. the arcus cosine of a rational number.
By Niven's theorem this is not a rational multiple of $\pi$.