Answer
$C=60{}^\circ,\ a\approx 34.5,$ and $b\approx 19.9$
Work Step by Step
First we will find the value of C
Properties of a triangle:
Sum of three angles is $A+B+C=180{}^\circ $
$\begin{align}
& A+B+C=180{}^\circ \\
& 85{}^\circ +35{}^\circ +C=180{}^\circ \\
& C=180{}^\circ -120{}^\circ \\
& C=60{}^\circ
\end{align}$
Now, we will find the remaining sides, using the ratio
$\frac{c}{\sin C}$ or $\frac{30}{\sin 60{}^\circ }$.
Now, we will use the law of sines to find a.
$\begin{align}
& \frac{a}{\sin A}=\frac{c}{\sin C} \\
& \frac{a}{\sin 85{}^\circ }=\frac{30}{\sin 60{}^\circ } \\
& a=\frac{30\sin 85{}^\circ }{\sin 60{}^\circ } \\
& a\approx 34.5
\end{align}$
Using the law of sines again we will find b.
$\begin{align}
& \frac{b}{\sin B}=\frac{c}{\sin C} \\
& \frac{b}{\sin 35{}^\circ }=\frac{30}{\sin 60{}^\circ } \\
& b=\frac{30\sin 35{}^\circ }{\sin 60{}^\circ } \\
& b\approx 19.9
\end{align}$
The solution is $C=60{}^\circ,\ a\approx 34.5,$ and $b\approx 19.9$.
