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You can use Snell's Law. Note that you need two angles for this - the incoming angle, and the outgoing angle.
The classical problem of angle trisection cannot be solved. If it were possible, it would provide the solution to a cubic equation. (-but it isn't and it won't!)
It depends on what angle. The main angles of a rectangle are all 90 degrees and knowing that does not help. If however, the angle in question is the angle made by the diagonal with one of the sides, then it is possible to use a basic trigonometric ratio to work out the length. Details will depend on which angle is given.
you subtract the other angle from 180 and that should give you your answers
Use the equation 180(n-2)/n, where n is the number of sides in the figure.