The measure of a given angle is180o
n, where n is a positive integer
not divisible by 3. Prove that this angle can be trisected by Eucliden means
(straightedge and compass).
It is possible to construct a regular pentagon whose angles are 72 and an equilateral triangles with angles 60. So it is possible to construct 72 - 60 = 12degrees. Bisect this angle with compass to get that 6 is constructible. Bisect again to show that 3 is constructible. However, 1 is not constructible because 3 cannot be trisected*. Also 2 cannot be constructed for that will imply that 1 (after bisection) can be constructed a contradiction. Thus, the smallest contructible angle is 3 degrees. Then any multiple of this can be constructed. Since $\displaystyle 180/n$ does not contain a factor of 3 it means after this division this integer is a multiple of three. Thus, it can be constructed. Q.E.D.
*)Because constructible numbers are algebraic over $\displaystyle \mathbb{Q}$ of degree power of two. Note that $\displaystyle 4\cos^3 x - 3\cos x - \cos 3x = 0$ showing that this polynomial is of degree 3 over the rationals.