A necessary and sufficient condition is established for the strict inequality pc(G∗)<pc(G) between the critical probabilities of site percolation on a quasi-transitive, plane graph G and on its matching graph G∗. It is assumed that G is properly embedded in either the Euclidean or the hyperbolic plane. When G is transitive, strict inequality holds if and only if G is not a triangulation. The basic approach is the standard method of enhancements, but its implemention has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity. This result is complementary to the work of the authors ("Hyperbolic site percolation", arXiv:2203.00981) on the equality pu(G)+pc(G∗)=1, where pu is the critical probability for the existence of a unique infinite open cluster. It implies for transitive G that pu(G)+pc(G)≥1, with equality if and only if G is a triangulation.