How to show that every compact 2-manifold, which is complex, is differentiable manifold?

Question

I'm struggling with this problem(Kahn, p.137).

Kahn gives some hints : (1) Prove that there is a cover of open sets, which overlap nicely, using the representation as a quotient of the disc, by identification on the boundary. (2) Show that a 2-manifold, which is differentiable, remains a differentiable manifold when a handle, or a cross-cap is attached.

I have no idea how these two hints are related.

Thank you.