A good deal of the geometry of a smooth irreducible subvariety of a complex abelian variety depends on ``how ample'' its normal bundle is. We show that a notion of non-degeneracy due to Ran is a good substitute for ampleness of the normal bundle for arbitrary (singular) irreducible subvarieties. Our main result is a Zak-type result for arbitrary subvarieties $V$ of an abelian variety that relates the dimension of the ``secant variety'' (defined as being $V-V$) to that of the ``tangential variety'' (defined in the smooth case as the union of the projectivized tangent spaces to $V$, translated at the origin). Corollaries include a new proof of the finiteness of the Gauss map and an estimate on the ampleness of the normal bundle of a smooth non-degenerate subvariety.