A

在$\triangle BDE$和$\triangle CFD$中，$\left\{\begin{array}{l}{BE=CD}\\{∠B=∠C}\\{BD=CF}\end{array}\right.$，
$\therefore \triangle BDE$≌$\triangle CFD$，
$\therefore \angle BED=\angle CDF$，
$\because \angle A+\angle B+\angle C=180^{\circ}$，
$\therefore \angle B=\frac{180°-∠A}{2}$，
$\because \angle BDE+\angle EDF+\angle CDF=180^{\circ}$，
$\therefore 180^{\circ}-\angle B-\angle BED+a+\angle CDF=180^{\circ}$，
$\therefore \angle B=a$，
即$\frac{180°-∠A}{2}=a$，
整理得$2a+\angle A=180^{\circ}$.
故选：$A$.