Answer
35 diagonals.
Work Step by Step
We must select two vertices among ten vertices. The order in which they are selected is not important. So, we have $_{10}C_2$ different combinations:
$_{10}C_2=\frac{10!}{(10-2)!~2!}=\frac{10\times9\times8!}{8!~2!}=45$
But, among those 45 combinations, 10 are sides.
$Diagonals=45-10=35$
