Quadrilateral $PQRS$ is inscribed in a circle. If $\angle P = (2x + 10)^{\circ}$ and $\angle R = (3x - 20)^{\circ}$, find the value of $x$ and the measures of $\angle P$ and $\angle R$.

Opposite angles of a cyclic quadrilateral sum to $180^\circ$.
$$\angle P + \angle R = 180^{\circ}$$ $$(2x + 10) + (3x - 20) = 180$$ $$\implies 5x - 10 = 180$$ $$\implies 5x = 190$$ $$\implies x = 38$$

Now,
$$\angle P = 2(38) + 10 = 86^\circ$$ $$\angle R = 3(38) - 20$$ $$\quad = 114 - 20$$ $$\quad = 94^\circ$$