I am familiar with the famous Gauss-Wantzel Theorem which states that a polygon is constructible with compass and straightedge if and only its number of sides is a power of two multiplied by a product of distinct Fermat primes. This theorem then provides a list of constructible angles. My question is whether a similar theorem for all constructible angles exists, that is, a necessary and sufficient condition for an angle $\alpha$ to be constructible using only compass and straightedge. Are there other constructible angles (which of course would not be the interior angles of a polygon) which are not of the form implied by the Gauss-Wantzel Theorem,and if so, what do they have in common with one another?
If possible, I would greatly appreciate a link to the theorem and its proof.