  We consider the two-dimensional capillary-gravity water waves problem where
the free surface $\Gamma_t$ intersects the bottom $\Gamma_b$ at two contact
points. In our previous works \cite{MW2, MW3}, the local well-posedness for
this problem has been proved with the contact angles less than $\pi/16$. In
this paper, we study the case where the contact angles belong to $(0, \pi/2)$.
It involves much worse singularities generated from corresponding elliptic
systems, which have a strong influence on the regularities for the free surface
and the velocity field. Combining the theory of singularity decompositions for
elliptic problems with the structure of the water waves system, we obtain a
priori energy estimates. Based on these estimates, we also prove the local
well-posedness of the solutions in a geometric formulation.
