<BAX = <ADY
AB || CD Property of a square.
<BAX = Theta Naming an angle (doesn't need proof)
<BAX = <AZD Property of parallel lines
<ZDY = 90 - Theta Right angle triangle
<YDA = theta The two angles at D are complementary.
Which was what you were intended to prove.
Triangle DAY and ABX are Congruent.
You know this because you have two angles and a side that are equal. If two angles in a triangle are equal then the third angle is equal. You have AAS but you could easily make it into ASA. The key step is the one above.
Nice problem. Thanks for posting.