Two angles are called supplementary when their sum is π.
I propose that two non-empty subsets of Sn be called supplementary
when no point of one is within 90 degrees of a point in the other and neither can be extended without violating the rule.
Two supplementary subsets are each convex and are separated by an equator (which is an Sn–1).
Two supplementary subsets of S1 subtend supplementary angles.
The supplementary subset of a polyhedral subset is polyhedral.
Alas, the content of a subset does not determine the content of its supplement.
This makes generalizations of Gauss-Bonnet more complex.
A subset of Sn is convex if it is the intersection of some non-empty set of hemispheres.
The shorter geodesic path between two points in a convex set lies within that set.