In an influential 1964 article, P. Lax studied genuinely nonlinear 2 X 2 hyperbolic PDE systems (in one space dimension). He showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two space dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Our main result is that the Lax-type wave breaking is stable under small Sobolev-class perturbations of the data that break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of true null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations.