Bisect an Angle

The easiest way to bisect an angle is to make a kite.  A kite in geometry is a quadrilateral with two pairs of adjacent congruent sides.  The diagonal joining the vertices where the congruent sides meet each other bisects both angles formed by those sides:

In this kite, AB = AD and BC = DC.  Diagonal AD bisects angle BAD and BCD.

Suppose C is a given angle you want to bisect. 

First you mark off segments CB and CD with the compass set to any convenient length:

Then you locate point A so it is equidistant from B and D:

Finally draw the bisector, CA:

Here is an animation of this construction.

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