1. In $\triangle A B C$, the lengths of the sides opposite to $\angle A 、 \angle B 、 \angle C$ are $a 、 b 、 c$ respectively.
Proposition $p: \angle B+\angle C=2 \angle A$, and $b+c=2 a$;
Proposition $q: \triangle A B C$ is an equilateral triangle.
Then proposition $p$ is a ( ) condition for proposition $q$.
(A) Sufficient and necessary
(B) Sufficient but not necessary
(C) Necessary but not sufficient
(D) Neither sufficient nor necessary
- 1. A.

If proposition $q$ holds, obviously, proposition $p$ holds.
If proposition $p$ holds, then $\angle A=60^{\circ}$.
By the cosine rule, we have $a^{2}=b^{2}+c^{2}-b c$.
Thus, $4\left(b^{2}+c^{2}-b c\right)=(b+c)^{2}$
$$
\Rightarrow(b-c)^{2}=0 \Rightarrow b=c \text {. }
$$

Therefore, $\triangle A B C$ is an equilateral triangle, which means proposition $q$ holds. In summary, proposition $p$ is a sufficient and necessary condition for proposition $q$.