If a graph with $n$ vertices has more than $\frac{(n-1)(n-2)}{2}$ edges then it is connected.I am a bit confused about this question, since I can always prove that for a graph to connected you need more than $|E|>n-1$  edges.
Every simple undirected graph with more than $(n-1)(n-2)/2$ edges is connected
graph theory
I am not sure what bothers you but as I see it you are confused about the following two facts If a graph is connected then $e \geq n-1.$If a graph has more than $e > \frac{(n-1)(n-2)}{2}$ then it is connected.Notice that the implications in 1 and 2 are in opposite directions.For a proof of 2. you can check out this link.