2. Given $I$ as the incenter of $\triangle A B C$, connect $A I, B I, C I$. If one of the triangles $\triangle B I C, \triangle C I A, \triangle A I B$ is similar to $\triangle A B C$, find the sizes of the angles of $\triangle A B C$.
Solution: Assume without loss of generality that $\triangle B I C$ is similar to $\triangle A B C$, then $\angle B I C \neq \angle A$.
In fact, if $\angle B I C=\frac{1}{2} \angle A+90^{\circ}=\angle A$, then $\angle A$ $=180^{\circ}$, which is a contradiction.

Assume without loss of generality that $\angle I B C=\angle A, \angle B I C=\angle C, \angle I C B=$ $\angle B$, then we have $\frac{1}{2} \angle B=\angle A, 90^{\circ}+\frac{1}{2} \angle A=\angle C$, $\frac{1}{2} \angle C=\angle B$, thus $\angle A=\frac{\pi}{7}, \angle B=\frac{2 \pi}{7}, \angle C=\frac{4 \pi}{7}$.
Therefore, the angles of $\triangle A B C$ are $\frac{\pi}{7}, \frac{2 \pi}{7}, \frac{4 \pi}{7}$.