The ratio of the number of sides of two regular polygons is 2 : 3 and the ratio of the sum of their interior angles is 5 : 8. What is the number of sides of the first polygon and the measure of the interior angle (in degrees) of the second polygon, respectively?

Let the sides of the polygons be 2x and 3x. 

Hence, the ratio of the sum of interior angles for the two polygons is: $$\ \frac{\ \left(2x-2\right)\times\ 180}{\left(3x-2\right)\times\ 180}$$ = $$\ \frac{\ 5}{8}$$

On solving this, x=6

This increases the number of sides in the first polygon to 12 and 18 for the second one.

To find the measure of each interior angle of the second polygon, we get $$\ \frac{\ \left(18-2\right)\times\ 180}{18}$$, which is 160 degrees.