Answer
$C=47^{\circ}$
$a=26.7$
and
$b=64.3$
Work Step by Step
I. Need to find the last angle.
$23^{\circ}+110^{\circ}+\angle C=180^{\circ}$
$\implies \angle C=47^{\circ}$
II. Use law of sines formula to find $a$ and $b$
$\dfrac{\sin 23^{\circ}}{a}=\dfrac{\sin 110^{\circ} }{b}=\dfrac{\sin 47^{\circ} }{50}$
This implies that
$a=\dfrac{50 \sin 23^{\circ} }{\sin 47^{\circ}}=26.7$
and
$b=\dfrac{50 \sin 110^{\circ} }{\sin 47^{\circ}}=64.3$
