2. In $\triangle A B C$, $\angle A=60^{\circ}$, points $M$, $N$, and $K$ are on sides $B C$, $A C$, and $A B$ respectively, such that $B K=K M=M N=$ $N C$. If $A N=2 A K$, find the measures of $\angle B$ and $\angle C$.
2. As shown in Figure 6, take the midpoint $P$ of $AN$. Then $AK = AP = PN$. Therefore, $\triangle APK$ is an equilateral triangle.
Hence $\angle ANK$
$$
=\frac{1}{2} \angle KPA = 30^{\circ} \text{.}
$$

Let $\angle ACB = \angle NMC = \alpha$. Then
$$
\angle ABC = \angle KMB = 120^{\circ} - \alpha \text{.}
$$

Therefore, $\angle KMN = 60^{\circ}$, which means $\triangle KMN$ is an equilateral triangle.
Thus, $\angle ANM = 90^{\circ}, \alpha = 45^{\circ}$.
Hence $\angle C = 45^{\circ}, \angle B = 75^{\circ}$.