Explanation:
Set the compass to some convenient radius. Use the same setting for all arcs and marks described below.
Using the vertex of the angle as the center, put marks on each leg of the angle. Call those marks A and B.
Using A as a center, draw an arc in the area where you believe the angle bisector will be located. If you like, you don't even need to guess. You can just draw a circle with A as the center.
Repeat the above instruction using B as the center. The arcs (or circles) will intersect. Call the point of intersection C. The angle bisector is the line from the vertex through point C.