5. In $\triangle A B C$, $A B=1, B C=2$. Then the range of $\angle C$ is ( ).
(A) $\left(0, \frac{\pi}{6}\right]$
(B) $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
(C) $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$
(D) $\left(0, \frac{\pi}{2}\right)$
5. A.

It is known that $A B=c=1, B C=a=2$.
By the Law of Sines, we have
$$
\sin C=\frac{c}{a} \sin A=\frac{1}{2} \sin A \in\left(0, \frac{1}{2}\right] \text {. }
$$

Therefore, $\angle C \in\left(0, \frac{\pi}{6}\right]$.