HMMT 2023 T3

Let $A B C D$ be a convex quadrilateral such that $\angle A B C=\angle B C D=\theta$ for some angle $\theta<90^{\circ}$. Point $X$ lies inside the quadrilateral such that $\angle X A D=\angle X D A=90^{\circ}-\theta$. Prove that $B X=X C$.