Angle chasing problem with quadrilateral - possible approaches?

Question

CONTEXT

I was recently working on some Langley-style problems, and wanted to construct some others, based on the "reverse engineering" approach I developed to solve them.

PROBLEM

The quadrilateral $ABCD$ is such that $\measuredangle ABC = 51^\circ$ and $\measuredangle DAB = 135^\circ$. Furthermore, a point $P$ inside the quadrilateral and a point $Q\in BC$ can be found such that

1. $AP \parallel DQ$,
2. $AD \cong AP \cong PQ$,
3. $CD \cong PD \cong PC$.

What is $\measuredangle BCD$?

QUESTIONS

I have a solution, since I myself constructed the problem (should I give it here? Or would this just be some sort of "bias"?). But I am wondering

1. Are there feasible/interesting approaches to solve the problem?
2. Are there even more than one solution to it?

EDIT To avoid any confusion. This is not a test or a homework. I am not even a student. My solution is $\measuredangle BCD= 78^\circ$, but I obtained it from a cumbersome geometric construction, and I am wondering if this can be obtained in some more direct way, and if there are other solutions.