  We consider the Cauchy problem for the 2D gravity water wave equation.
Recently Wu \cite{Wu15, Wu18} proved the local well-posedness of the equation
in a regime which allows interfaces with angled crests as initial data. In this
work we study properties of these singular solutions and prove that the
singularities of these solutions are "rigid". More precisely we prove that an
initial interface with angled crests remains angled crested, the Euler equation
holds point-wise even on the boundary, the particle at the tip stays at the
tip, the acceleration at the tip is the one due to gravity and the angle of the
crest does not change nor does it tilt. We also show that the existence result
of Wu \cite{Wu15} applies not only to interfaces with angled crests, but also
allows certain types of cusps.
