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 ½n(n – 3), so we also know:
½m(m – 3) + ½n(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.