Diagonals of Two Regular Polygons

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?


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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.

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