The number of diagonals of two convex polygons together is 158, and the sum of their angles is $4320^{\circ}$. How many sides does each polygon have?
For the number of diagonals and the sum of interior angles of polygons, we can write two equations using the relevant formulas.

Let $n_{1}$ be the number of sides of one polygon, and $n_{2}$ be the number of sides of the other polygon.

Then

$$
\begin{gathered}
\frac{n_{1}\left(n_{1}-3\right)}{2}+\frac{n_{2}\left(n_{2}-3\right)}{2}=158 \\
\left(n_{1}-2\right) 180^{\circ}+\left(n_{2}-2\right) 180^{\circ}=4320^{\circ}
\end{gathered}
$$

The solution to the system of equations is 16 and 12. Of course, solving the problem does not end here. We must verify that the sum of the diagonals and the sum of the interior angles in a 16-sided and a 12-sided polygon indeed match the given values.