Summary: In this paper we obtain a tangency principle for hypersurfaces of an arbitrary Riemannian manifold, with not necessarily constant \(r\)-mean curvature function \(H_r \). More precisely, we obtain sufficient geometric conditions for two submanifolds of a Riemannian manifold to coincide, as a set, in a neighborhood of a tangency point. As applications of our tangency principle, we obtain, under certain conditions on the function \(H_r\), sharp estimates on the size of the greatest ball that fits inside a connected compact hypersurface embedded in a space form of constant sectional curvature \(c\leq 0\) and on the size of the smallest ball that encloses the image of an immersion of a compact Riemannian manifold into a Riemannian manifold with sectional curvatures limited from above. This generalizes results of Koutroufiotis, Coghlan-Itokawa, Pui-Fai Leung, Vlachos and Markvorsen. We also generalize a result of Serrin. Our techniques permit us to extend results of Hounie-Leite.