jmc

Let $x$ be the degree measure of $\angle A$. Then the degree measures of angles $B$, $C$, and $D$ are $x/2$, $x/3$, and $x/4$, respectively. The degree measures of the four angles have a sum of 360, so $$360=x+\frac{x}{2}+\frac{x}{3}+\frac{x}{4}=\frac{25x}{12}.$$Thus $x=(12\cdot 360)/25=172.8\approx \boxed{173}$.