HMMT 2011 U20

Let $A B C D$ be a quadrilateral circumscribed about a circle with center $O$. Let $O_{1}, O_{2}, O_{3}$, and $O_{4}$ denote the circumcenters of $\triangle A O B, \triangle B O C, \triangle C O D$, and $\triangle D O A$. If $\angle A=120^{\circ}, \angle B=80^{\circ}$, and $\angle C=45^{\circ}$, what is the acute angle formed by the two lines passing through $O_{1} O_{3}$ and $O_{2} O_{4}$ ?