Answer
The required values are $A=44{}^\circ,b\approx 18.6$, and $c\approx 22.8$.
Work Step by Step
At first we will find A.
Property of a triangle:
Sum of three angles is $A+B+C=180{}^\circ $
$\begin{align}
& A+B+C=180{}^\circ \\
& A+54{}^\circ +82{}^\circ =180{}^\circ \\
& A=180{}^\circ -136{}^\circ \\
& A=44{}^\circ
\end{align}$
Now, to find the remaining sides we will use the ratio $\frac{a}{\sin A}$ or $\frac{16}{\sin 44{}^\circ }$,
Now, we will use the law of sines to find b.
$\begin{align}
& \frac{b}{\sin B}=\frac{a}{\sin A} \\
& \frac{b}{\sin 54{}^\circ }=\frac{16}{\sin 44{}^\circ } \\
& b=\frac{16\sin 54{}^\circ }{\sin 44{}^\circ } \\
& b=18.6
\end{align}$.
Again use the law of sines to find c
$\begin{align}
& \frac{c}{\sin C}=\frac{a}{\sin A} \\
& \frac{c}{\sin 82{}^\circ }=\frac{16}{\sin 44{}^\circ } \\
& c=\frac{16\sin 82{}^\circ }{\sin 44{}^\circ } \\
& c=22.8
\end{align}$
The solution is $A=44{}^\circ,b\approx 18.6$, and $c\approx 22.8$.
