Draw the angle.
Place the compass point on the vertex.
Draw an arc that intersects both sides of the angle.
Without changing the compass width, place the compass point on one intersection point.
Draw an arc inside the angle.
Without changing the compass width, place the compass point on the other intersection point.
Draw another arc so it intersects the first arc.
Draw a straight line from the vertex through the intersection of the two arcs.
This line is the angle bisector.