We show that there is a matching between the edges of any two triangul
ations of a planar point set such that an edge of one triangulation is
matched either to the identical edge in the other triangulation or to
an edge that crosses it. This theorem also holds for the triangles of
the triangulations and in general independence systems. As an applica
tion, we give some lower bounds for the minimum-weight triangulation w
hich can be computed in polynomial time by matching and network-flow t
echniques. We exhibit an easy-to-recognize class of point sets for whi
ch the minimum-weight triangulation coincides with the greedy triangul
ation.