The total number of sides in two regular polygons is 13, and total number of diagonals is 25. How many sides are in each polygon?

Let x be the sides of one polygon

y be the sides of the other polygon

x + y = 13

y = 13 - x

then the formula for the number of polygons is n (n - 3) / 2

x (x - 3) / 2 + y (y - 3) / 2 = 25

x (x - 3) / 2 + (13 - x) (13 - x - 3) / 2 = 25

x (x - 3) / 2 + (13 - x) (10 - x) / 2 = 25

solve for x

x = 8

y = 13 - 8 = 5