We need to be able to prove this question.
(ITT) Given a triangle with two if its sides congruent, then are the two angles opposite those sides also congruent? (Look at this on a plane, sphere, and hyperbolic plane.)
I already know that I need to use symmetries to solve this problem. I need to then look at the plane and note what properties of a plane I can use. Then I need to look at other surfaces and try to find counterexamples.
If I think that ITT is not true for all triangles on a particular surface then I need to describe a counterexample and look for a smaller class of triangles that do satisfy ITT on that surface.
I need to state explicitely what properties I will be using.
A hint to this problem is this:
On a sphere two given points do not determine a unique geodesic segment but two given points plus a third point collinear to the given two do determine a unique geodesic segment.
Any help on this problem will be very appreciated!