The author gives a new integral estimate of distances and angles on manifolds with a given lower Ricci curvature bound. He obtains this estimate via a Hessian estimate and states it in three different forms. Using this, he proves (among other things) the following conjectures: 1. The volume is a continuous function on the space of all closed \(n\)-manifolds with Ricci curvature greater or equal to \(-(n-1)\) equipped with the Gromov-Hausdorff metric (a conjecture of Anderson and Cheeger). 2. An almost nonnegatively Ricci curved \(n\)-manifold with first Betti number equal to \(n\) is a torus (Gromov's conjecture). 3. An open \(n\)-manifold with nonnegative Ricci curvature, whose tangent cone at infinity is \(\mathbb{R}^n\), is itself \(\mathbb{R}^n\) (a conjecture of Anderson and Cheeger).