You mean DY? It seems that the situation you describe can arise only if Y is outside the quadrilateral.
There is given convex quadrilateral ABCD. And internal bisectors of angle A and C intersect in point X. And internal bisectors of angle B and D intersect in point Y. And XAY = 90 . Prove that XCY = 90. I've noticed that AY is the external bisector of A. Please help me !
http://i.stack.imgur.com/tX0ue.png)