How to prove that a quadrilateral with the following properties is a parallelogram?

Question

Suppose that there is a quadrilateral $ABCD$.

1. Diagonal $\overline{BD}$ bisects $\overline{AC}$ (but it is not said that $\overline{AB}$ bisects $\overline{BD}$). If $O$ is a point of intersection of $\overline{AC}$ and $\overline{BD}$ then $\overline{AO}=\overline{OC}$.
2. Angle $\hat{A}$ is equal to angle $\hat{C}$

Prove that $ABCD$ is a parallelogram.

I don't know how to start, because the side $\overline{BD}$ intersects equal angles and I can't find similar triangles.

Answer 1

From what's described so far, it can't be proven, as the quadrilateral could be a kite, like this