To bisect an angle with a compass, draw an arc from the angle's vertex that intersects the two lines coming out. Then draw two more arcs from where the first arcs intersected with the angle's two lines. These arcs will form a third intersection. A straight line that passes through the vertex and the third intersection is the angle bisector.
The proof that this method works was given as a proposition in Euclid's "Elements Book 1". Euclid's "Elements" is a five-book treatise that proved everything known about geometry at the time that it was written.
Dividing an angle into three even angles using only a pencil, compass and straight edge may sound very similar to bisecting, but in 1836 the mathematician Pierre Wantzel proved that no general method of trisection exists.