Pythagorean triples: for example, 3^2+4^2=5^2. For a length-6 plate the distance from the first to the last stud is 5, so you can place it at an angle of approximately 25 degrees, i.e. arctan 4/3, as long as there are no studs in the way (which you can achieve easily using tiles). On LDD you can place it horizontally, choose the rotate tool, and the constraint engine will automatically forbid invalid angles around arctan 4/3, where the plate would overlap with the stud, and "snap" to the right one.

An example of a rotation is to build two rectangles of the same size, and then place the topleft-bottomright diagonal of one rectangle on top of the topright-bottomleft diagonal of the other. The diagonals don't have an integral length, but the lengths are the same so it just works.