I'm not sure if I understand this question right.
Q. Given an angle at a point A and given another segment BC, construct a point D so that the angle DBC equals the given angle at A.
Figure - 
Should I assume one of the sides of the angle a is bc and the end point of the other side is the point d ?
David Mitra was spot on. Go the the following link it has the proof and a java video to help you understand. Are you in Math430 UMD?
http://www.mathopenref.com/constcopyangle.html