We prove central limit theorems and establish rates of convergence for the following problems in geometrical probability when points are generated in the $\lbrack 0,1\rbrack^2$ cube according to a Poisson point process with parameter $n$: 1. The length of the nearest graph $N_{k,n}$, in which each point is connected to its $k$th nearest neighbor. 2. The length of the Delaunay triangulation $\operatorname{Del}_n$ of the points. 3. The length of the Voronoi diagram $\operatorname{Vor}_n$ of the points. Using the technique of dependency graphs of Baldi and Rinott, we show that the dependence range in all these problems converges quickly to 0 with high probability. Our approach also establishes rates of convergence for the number of points in the convex hull and the area outside the convex hull for points generated according to a Poisson point process in a circle.