Concerning Triangles

Triangles, as with lines, vary between types of geometry.  This is merely an overview of their behavior on different planes. A more detailed analysis of triangle similarity and non-Euclidean trigonometry is in store as soon as I figure out how, exactly, non-Euclidean trigonometry works.
Euclidean geometry: Triangle angles always add to 180 degrees.

Elliptic geometry: Triangle angles always add to more than 180 degrees. The angles of smaller triangles will add to only a little more than 180 degrees; for instance, the amount over 180 degrees the angles of a triangle drawn on a patch of dirt would be infinitesimal, but the disparity becomes more pronounced with a shape as large as the Bermuda triangle. Angles will never add to more than 900 degrees. Triangles are only similar when congruent, because a larger triangle will have larger angle measures.

Hyperbolic geometry: Triangle angles add to less then 180 degrees, but always to more than 0 degrees, with larger triangles being close to 0. Similarly to triangles on elliptic planes, they are only similar when congruent.

Figure 1 (by Hugh Gray Lieber)