A right angle has 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?
An exterior angle of a triangle is the sum of the opposite interior angles. Therefore, the ratio of the interior angles of a triangle to the exterior angles of a triangle is 1:1. Since the measures of the exterior angles of our triangle are in the ratio 13:14, the measures of the interior angles are in the ratio $\frac{13}{13+\!14}:\frac{14}{13+\!14} = 13:14.$ Therefore, the smallest interior angle of the triangle is $\frac{13}{13+14} \cdot 180^\circ = \boxed{90^\circ}.$
