Answer
$B =60^{\circ}$, $a = 5.77$ and $c=11.54$
Work Step by Step
We have: $B=180^{\circ}-30^{\circ}-90^{\circ}=60^{\circ}$
Use Law of Sines to find side $a$:
$\dfrac{\sin B}{b}=\dfrac{\sin A}{a}$
This implies that $a=\dfrac{10 \sin 30^{\circ}}{\sin 60^{\circ}} \implies a \approx 5.77^{\circ}$
Use Law of Sines to find side $c$:
$\dfrac{\sin A}{a}=\dfrac{\sin C}{c}$
This implies that $c=\dfrac{5.77 \sin 90^{\circ}}{\sin 30^{\circ}} \implies a \approx 5.77^{\circ}$
Our answers are: $B =60^{\circ}$, $a = 5.77$ and $c=11.54$
