Another interesting problem!
Suppose you have a quadrilateral ABCD where AB is parallel to CD and such that the length of DA = AB = BC and BD = DC.
What are the angles of this quadrilateral?
Hint: Obviously, a trigonometry solution is possible with some work. But there’s another way, without writing down a single number! Keep in mind your shapes and rules about certain, specific edges being equal to one another as in this problem.
Mess or Math? 
Comments on: "Clever Geometry Problem" (3)
A pretty interesting problem. Thanks for this. I tried doing a quick sketch of the solution. Hopefully, there’s no flaw. The shape is a parallelogram with angles,
and $\latex {2 \pi}{3}$
Sorry, I meant, and $\latex \frac {2{\pi}}{3}$
Thank you for your interest in the problem! This is a pretty interesting problem because I believe that there are a couple of answers. Try and see if you can get another solution!