In two-dimensional systems, predictability and visibility satisfy a complementarity relation, which has been generalized to bipartite systems by including the concurrence. We consider systems in which the two alternatives of a which-state experiment are nonorthogonal states and propose generalized measures of predictability and visibility. Criteria like symmetry under exchange of the alternatives, vanishing visibility when only one of the alternatives is present, and clear operational meaning, favor definitions of predictability and visibility which use a three-dimensional basis, previously employed in the unambiguous discrimination of nonorthogonal states. We derive generalized single-partite and bipartite complementarity relations and find that predictability and visibility do not exhaust the single-partite properties for a nonorthogonal basis. A recent experiment on the quantum-classical border is analyzed using generalized complementarity relations.