Actually, it occurs to me that I've never actually sat down and checked that angle addition obeys the triangle inequality in 3 dimensions -- I suppose if nothing else it can be done by lots of inequality grinding

The ordinary triangle inequality is immediate from -- is practically identical to -- the statement that a straight line is the shortest distance between two points.

The spherical triangle inequality, in any number of dimensions, is the same thing with "straight line" replaced by "great circle". A detail that doesn't arise for flat space is that there are two angles between two lines (whereas there is only one distance between two points in flat space), and you have to choose one that is no more than pi.