  We prove a quadratic interaction estimate for wavefront approximate solutions
to the triangular system of conservation laws \[ \begin{cases} u_t + \tilde
f(u,v)_x = 0, \cr v_t - v_x = 0. \end{cases} \] This quadratic estimate has
been used in the literature to prove the convergence rate of the Glimm scheme.
  Our aim is to extend the analysis, done for scalar conservation laws, in the
presence of transversal interactions among wavefronts of different families.
The proof is based on the introduction of a quadratic functional $\mathfrak
Q(t)$, decreasing at every interaction, and such that its total variation in
time is bounded. %cancellations it variation is controlled by the total
variation growths at most of the total variation of the solution multiplied by
the amount of cancellation.
  The study of this particular system is a key step in the proof of the
quadratic interaction estimate for general systems: it requires a deep analysis
of the wave structure of the solution $(u(t,x),v(t,x))$ and the reconstruction
of the past history of each wavefront involved in an interaction.
