A right triangle 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?
The sum of the angle measures in a triangle is 180 degrees. So the two exterior angles in question have measures (13/27)(180) and (14/27)(180) degrees, so their measures are 90 + (13/15)(90) and 90 + (14/15)(90) degrees. Since these two exterior angles are the two acute angles of the right triangle, the sum of their measures must be 90 degrees, or
$90 + \frac{13}{15} \cdot 90 + 90 + \frac{14}{15} \cdot 90 = 270.$
Since these are the measures of the acute angles of the triangle, the measures of the three angles of the triangle are 90, 90 + (13/15)(90), and 90 + (14/15)(90) degrees. The measure of the smallest angle is $\boxed{90}$ degrees.