Question 8491
 Assume it is a convex polygon with n sides

 For each vertex v,  there are n-3 possible diagonals passing through v.
 (except the two adjacent vertices & v itself)
 
 Since a diagonal (a line segment was counted twice from the two end
 vertices), totally there are n(n-3)/2 = 35 diagonals.

 Solve n(n-3)= 70, or n^2 - 3n - 70 = 0
 Factoring (n-10)(n+7) = 0.
 So n = 10 or n = -7(Illegal)

 Answer : n = 10

 Forget another terribly wrong answer,

 Kenny