2. In $\triangle A B C$, it is known that $A B=a^{2}, B C=4 a$, $A C=b^{2}-4$, and $a, b$ are odd numbers greater than 3. Then the relationship between $a$ and $b$ is ( ).
(A) $a>b$
(B) $a<b$
(C) $a=b$
(D) cannot be determined
2. C.

It is known that $a^{2}>4 a$. By the triangle inequality, we have
$$
\begin{array}{l}
a^{2}-4 a<b^{2}-4<a^{2}+4 a \\
\Rightarrow(a-2)^{2}<b^{2}<(a+2)^{2} \\
\Rightarrow a-2<b<a+2 .
\end{array}
$$

Since $a$ and $b$ are positive odd numbers, we have $b=a$.