to speak of the sum of angles in a triangle in a geometry you just need some way to compare angles, AND ADD THEM AND THEN TO SAY THAT THE SUM OF ANGLES IN A TRIANGLE IS MORE THAN (oops) a straight angle.
if you want to prove such examples exist, it depends what your standards of belief are. if you are someone who believes only in euclidean space, then for you it is necessary to find every other example embedded there.
it is quite consistent to imagine spherical or hyperbolic geometries, where triangles add to other than a straight angle, but to produce examples of them, we often look in euclidean space of higher dimension.
i think Halls meant that for a triangle with angle sum more than a straight angle, we probably look in a spherical geometry, and for less, in a hyperbolic geometry.