2. In $\triangle A B C$, it is known that $\angle A C B=45^{\circ}, D$ is any point on side $A B$ other than points $A$ and $B$, and the circumcenters of $\triangle A B C$, $\triangle A D C$, and $\triangle B D C$ are $O$, $O_{1}$, and $O_{2}$, respectively. Then the degree measure of $\angle O_{1} O O_{2}$ is $\qquad$.
2. $135^{\circ}$.

As shown in Figure 3, let the lines $O O_{1}$ and $O O_{2}$ intersect $A C$ and $B C$ at points $E$ and $F$, respectively. Then
$$
\begin{array}{l}
\mathrm{OO}_{1} \perp A C, \\
\mathrm{OO}_{2} \perp B C .
\end{array}
$$

Thus, points $C$, $E$, $O$, and $F$ are concyclic.
$$
\begin{array}{l}
\text { Therefore, } \angle O_{1} O O_{2} \\
=180^{\circ}-\angle A C B=135^{\circ} \text {. }
\end{array}
$$