解：AF是圆O的直径，五边形ABCDE是圆O的内接正五边形， $\therefore \widehat{C F}=\widehat{D F},\widehat{B C}=\widehat{D E},\angle B A E=108^{\circ}$,

$\therefore \widehat{B F}=\widehat{E F}$，

$\therefore \angle B A F=\frac{1}{2} \angle B A E=54^{\circ}$，

$\therefore \angle B D F=\angle B A F=54^{\circ}$.