Do I need a 10-24mm f/3.5-4.5 wide angle lens?

I was wondering, I have a nikkor 18-55mm f/3.5-5.6 and a nikon d3100 do I need a 10-24mm f/3.5-4.5 wide angle lens to do wide angle photography

Only you know if 18mm is not wide enough for what you want to do.

No. If you want to shoot wider than 18mm you do but it's not necessary.

18mm is already very wide and it doesn't distort the image significantly unlike those with shorter focal lengths.

Well 10mm is considerably wider than 18mm.

You don't "need" it, as 18mm is already wide angle, but if you want to take really wide angle shots, go for it. Extreme wide angle shots can add quite a lot of distortion to an image, but it can look very dramatic for a landscape. The choice is entirely yours.

Can't answer the question because it's basically unanswerable. You are asking us to get inside your head and decide what you want for yourself. This is just silly. Make your own mind up.