We have considered the theory of breakdown of an arbitrary gas-dynamic discontinuity for the space-time dimension equal to two. The link of this task with the geometrical theory of reconfiguration of shock-waves and wave fronts is shown. We consider the Riemann problem of the breakdown of an arbitrary discontinuity of parameters at angular collision of two flat flows. The problem is solved as accurate stated. We consider the solution region with different types of the shock-wave structure. The Mach number region is discovered and the angles of flows interaction for which there is no solution. We demonstrate the generality of solutions for one-dimensional non-stationary and two-dimensional stationary cases.