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

We have a regular *m*-gon and a regular *n*-gon where *m* + *n* = 13. The number of distinct diagonals in an *n*-gon is ½*n*(*n* – 3), so we also know:

½

*m*(*m*– 3) + ½*n*(*n*– 3) = 25.Substituting *m* = 13 – *n*, and solving the resulting quadratic equation, we get *n* = 5 or 8. So we have a pentagon (5 angles) and an octagon (8 angles).

Check: 5 + 8 = 13 and ½(5)(5 – 3) + ½(8)(8 – 3) = 5 + 20 = 25.