As was already noted in comments, the problem admits no solution using compass and straight edge. You can render the problem solvable by either allowing neusis or tomahawk.
Here I will explain the Archimedes solution using neusis.
!neusis construction
Given an angle $\alpha$, draw a circle centered at its tip point $\mathbf{O}$. Draw a chord $\mathbf{AC}$. Let $\beta = \angle \mathbf{BOC}$, and let $\gamma = \angle \mathbf{BCO}$.
It follows elementary that $\angle \mathbf{OBA} = \beta+\gamma$ and $\angle \mathbf{OAB} = \alpha - \gamma$. Since $\mathbf{OA} = \mathbf{OB}$ as radii, $ \alpha - \gamma = \beta+\gamma$, giving $\gamma = \frac{\alpha - \beta}{2}$. If we further impose $\beta = \gamma$, we get $\beta=\gamma=\frac{\alpha}{3}$.
In this configuration, $\mathbf{CB} = \mathbf{OB}$ as sides opposite to equal angles, which is how the neusis comes in.
One would use the marked ruler, to make $\mathbf{CB}$ equal to the radius of the circle.