Let the angle we wish to bisect be defined as follows.
We now use the compass to draw an arc which crosses both rays. We put the point of the compass on the angle's vertex.
Let's place the compass' point at the intersection between the arc and one of the rays and we draw a new arc between the rays.
We keep the legs of the compass fixed in the same angle and move its point to the intersection between the first arc and the other ray. We draw a new arc between the rays. This arc must intersect the arc we drew between the rays.
We now use a straightedge to draw a ray from the vertex through the intersection between the two arcs.
We will now remove the excess marks.
We are done. We have now bisected the angle.