What polygon has 35 diagonals?

A decagon.

Proof:

In an n sided polygon, each vertex would be have (n-3) diagonals attached to it, as it would be connected to every vertex other than itself and the two next to it by a diagonal. There are n sides, so there are n(n-3) ends of diagonals. Therefore there are (n(n-3))/2 diagonals in the polygon.

Taking the number of diagonals to be 35, we have:

(n(n-3))/2 = 35

n(n-3) = 70

which gives the quadratic

n2-3n-70 = 0

Solving this gives n = 10 and -7. -7 can be ignored, so the answer is 10.