Answer
The required values are $C=95{}^\circ,b\approx 81.0,$ and $c\approx 134.1$.
Work Step by Step
At first we will find C.
Property of a triangle:
The sum of three angles is $A+B+C=180{}^\circ $
$\begin{align}
& A+B+C=180{}^\circ \\
& 48{}^\circ +37{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -85{}^\circ \\
& C=95{}^\circ
\end{align}$
Now, to find the remaining sides, we will use the ratio $\frac{a}{\sin A}$ or $\frac{100}{\sin 48{}^\circ }$,
Now, we will use the law of sines to find b.
$\begin{align}
& \frac{b}{\sin B}=\frac{a}{\sin A} \\
& \frac{b}{\sin 37{}^\circ }=\frac{100}{\sin 48{}^\circ } \\
& b=\frac{100\sin 37{}^\circ }{\sin 48{}^\circ } \\
& b=81.0
\end{align}$
Again use the law of sines to find c
$\begin{align}
& \frac{c}{\sin C}=\frac{a}{\sin A} \\
& \frac{c}{\sin 95{}^\circ }=\frac{100}{\sin 48{}^\circ } \\
& c=\frac{16\sin 95{}^\circ }{\sin 48{}^\circ } \\
& c=134.1
\end{align}$
The solution is $C=95{}^\circ,b\approx 81.0$, and $c\approx 134.1$.
