A right angle ahs exterior angles at each of its acute angles with measures in the ratio 13:14. What is the measure of the smallest interior angle of the triangle?
Note that the exterior angles of a triangle sum to $360^\circ$. Since the two acute angles are in the ratio $13:14$, their measures must be $13x$ and $14x$ for some $x$. Since they are exterior angles, the measures of the corresponding interior angles are $180^\circ - 13x$ and $180^\circ - 14x$.
Since the angles of a right triangle sum to $180^\circ$, we get \begin{align*}
(180^\circ - 13x) + (180^\circ - 14x) + 90^\circ &= 360^\circ\\
360^\circ - 27x &= 360^\circ\\
x &= 0.
\end{align*}Therefore, the smallest interior angle is $180^\circ - 13x = \boxed{180^\circ}$.