I am having a hard time trying to come up with a definition of an angle on a sphere. Do the properties, such as for angles in a plane, apply to those in a sphere, and if so how I can prove they do (or do not) apply? Is it possible to create a square on a sphere?
I thought it was possible to create a sphere, just the angles would not have 90 degrees. I understand great circles are the "straight lines" on a sphere. Attempting to understand angles is hard though. Does ASS, SSA and other theorems which do not work in a plane, work on a sphere? Do the regular ones work on a sphere?
And in general on a plane, how do you prove that the opp. angles formed by 2 intersecting strght lines are congruent? What properties of straight lines are being used when trying to prove this? How are we able to check 2 angles are congruent (not using a protractor, ruler, etc, etc of course).
Some of my thoughts:
If you take a circle (not a great circle) on a sphere, it's analgous to a circle on a plane. Also, every great circle has TWO centers. Further, great circles are the only way to create an angle....
Angles in general have to have straight lines. We can think of angles on a sphere in respect to movement, dynamically, and geometric shape. Locally on a sphere, angles looks almost like a Euclidean space.
This stuff messes with my mind !