Thread: Isosceles Triangles

I am looking at the following surfaces: Euclidean plane, cylinder, cone, hyperbolic plane and the sphere.

On which of those surfaces can you have an isosceles triangle? I concluded that you can obviously have one in the plane and in a hyperbolic plane. However, on a sphere, you have to distinguish what your internal/external angles are, since 2 lines connect two points (shortest distance, then the one going around). As far as a cylinder and cone go, you can only have an isosceles triangle if it falls in one of the "leafs," (when you draw it out on paper). If it falls into more of them, you cannot.

My question: on those surfaces, if two angles of a triangle are congruent, will the sides opposite those angles ALWAYS be congruent as well? I can't think of a counter-example.

I've discovered one counter-example. On a cylinder, I can come up with a triangle that have two angles that are congruent, however, the sides opposite those angles are congruent, too. Similarly, I can find an example on the cone. I am unable to find a counter example on a hyperbolic plane or a sphere. On a sphere, one has to distinguish what the "internal" angles are.

Regarding the cylinder and cone, this is when it falls into more than one "leaf," that is, when drawing it in 2-D.