Answer: $\mathrm{C}$
Solution: When $\angle A=\angle C$ are both the smallest angles, at this time $\angle B$ reaches the maximum value $m^{\circ}$ among the largest angles.
$$
\angle A+\angle B+\angle C=180^{\circ} \text {, }
$$

which means
$$
\begin{array}{c}
2 \angle A+\angle B=180^{\circ}, \\
2 \times \frac{2}{5} \angle B+\angle B=180^{\circ},
\end{array}
$$

solving this gives
$$
\angle B=100^{\circ} \text {, }
$$

which means
$$
m=100 \text {. }
$$

When $\angle C=\angle B$ are both the largest angles, at this time $\angle B$ reaches the maximum value $n^{\circ}$ among the largest angles.
$$
\begin{array}{r}
\angle A+\angle B+\angle C=180^{\circ}, \quad \angle A+2 \angle B=180^{\circ}, \\
\frac{2}{5} \angle B+2 \angle B=180^{\circ},
\end{array}
$$

solving this gives
$$
\angle B=75^{\circ},
$$

which means
$$
n=75 \text {. }
$$

Therefore,
$$
m+n=175 .
$$

Hence, the answer is C.
23. In $\triangle A B C$, $\angle A$ is the smallest angle, $\angle B$ is the largest angle, and $2 \angle B=5 \angle A$. If the maximum value of $\angle B$ is $m^{\circ}$, and the minimum value of $\angle B$ is $n^{\circ}$, then $m+n=(\quad)$.
(A) 155
(B) 165
(C) 175
(D) 185