Construction of an Angle Bisector Using a Compass

Steps to construct the angle bisector of any angle (whose measure we do not know):
1. Draw ∠ABC of any measure.

2. Place the compass at point B and draw an arc to cut arms BA and BC. Name the points of intersection as P and Q, respectively.

3. Now, place the point of the compass at P and, taking a convenient distance, draw an arc inside the angle.  Using the same distance, draw another arc inside the angle from point Q to cut the previous one. 
4. Name the point of intersection as O. Now draw ray BO.  Ray BO is the bisector of ∠ABC.