The measures of the interior angles of a quadrilateral are in the ratio $1: 2: 3: 4$. Determine the measure in degrees of the smallest interior angle.
Let the measure of the smallest interior angle be $x^{\circ}$.

Then the other three interior angles have measures $2 x^{\circ}, 3 x^{\circ}$, and $4 x^{\circ}$.

The interior angles of a quadrilateral add to $360^{\circ}$, so $360^{\circ}=x^{\circ}+2 x^{\circ}+3 x^{\circ}+4 x^{\circ}=10 x^{\circ}$. Therefore, $360=10 x$, so $x=36$.

ANSWER: $36^{\circ}$

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