My guess:

Choose a point x along the edge of the surface of the liquid in the container, a point y along the rim of the container such that x and y are as far apart from each other as possible, and a point z on the edge of the surface of the liquid such that x and z are as far away as possible. Then, the angle between xy and xz is the largest possible angle at which the container can be held without spilling any of the contents inside. So, that angle plus whatever infinitesimally small amount would be the minimum angle required to tilt the glass such that a non-zero amount of liquid leaves the glass.

That's what seems right to me, but I can't back it up with anything other than "it seems right cause of the way it is."