The problem:
Suppose two convex pentagons $A$ and $B$ have equal interior angles (that is, $A=A_1A_2A_3A_4A_5$ and $B=B_1B_2B_3B_4B_5$) with $\angle A_j =\angle B_j$ for each $j\in\{1,\ldots,5\}$).
Suppose that $\mbox{int}(A) \approx \mbox{int}(B)$ are conformally equivalent with a biholomorphism $f:\mbox{int}(A) \rightarrow \mbox{int}(B)$ whose continuous extension to the boundary maps $A_i\overset{f}{\mapsto}B_i$.
Show that under these conditions, $A$ and $B$ are similar.
Ideas:
I would suspect the reflection principle would be applicable, but I'm not certain how to work out the proof.