$\angle \alpha=10^{\circ} 45^{\prime}$, 则 $\angle \alpha$ 的余角等于 \$ \qquad \$
$\because \angle \alpha=10^{\circ} 45^{\prime}$,

$\therefore \angle \alpha$ 的余角等于: $90^{\circ}-10^{\circ} 45^{\prime}=79^{\circ} 15^{\prime} ;$

故答案为: $79^{\circ} 15^{\prime}$.