You wouldn't (locally) be able to measure the angles you have shown on
figure. You would only be able to measure angles with respect to the local direction
of gravity (by, e.g., using a plumb-line). The surface of a spinnig self-gravitating
sphere, that deforms into an ellipsoid from the rotation, is well-approximated by a
equi-gravitational potential surface, which means the direction of gravity will be
perpendicular to that surface.
If you find the points on the respective surfaces where the local plumb-line points
in your five chosen directions, you'll find the statement in your quote to be true.
Paul Deema wrote: