  A necessary and sufficient condition is established for the strict inequality
$p_c(G_*)<p_c(G)$ 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 $p_u(G) + p_c(G_*) = 1$, where
$p_u$ is the critical probability for the existence of a unique infinite open
cluster. It implies for transitive $G$ that $p_u(G) + p_c(G) \ge 1$, with
equality if and only if $G$ is a triangulation.
