How to analyze triangles in Lobachevsky geometry? I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the general approach to this kind of task?
 A: Lobachevsky's geometry also goes by the more common name hyperbolic geometry.
I will assume that we are talking about the 2-dimensional case. There are two particularly convenient models: the Poincaré upper half plane and the Poincaré disc. The linked pages have plenty of identities between angles and sides in hyperbolic triangles. Many of the usual trigonometric identities hold (I imagine that you will have to prove that if you want to use it), but it is no longer true that angles of a triangle sum to $\pi$. Indeed, the sum is always strictly smaller (see formula for the area of a triangle).
A: There are a few models, and the great thing is that (once you've shown that they're equivalent) different things are easy to show in different models.  If you're clever about it, you can juggle back and forth and do as little work as possible!
As Alex mentions, there are the Poincare disc and UHP models.  Both of these "respect angles" (e.g. two hyperbolic geodesics are perpendicular iff they look perpendicular in either of these models).  There's another disc model which does not respect angles: hyperbolic geodesics are now just chords of the disc.  In this model, for instance, it's just a tad easier to prove that the postulate "two lines intersect in at most one point" holds than it is in either of the others (not that it's really so hard in any of them).  I'm sure there are less trivial things that are easiest to show in this model, but none are springing to mind.  But then, for instance, if you want to find a regular octagon whose angles are each 45 degrees (which comes up in topology as the fundamental domain of a 2-holed torus), you're going to be by far best off with the Poincare disc model.  As I said, each model has its own perks.
In my opinion, if you're dealing with right triangles it might be easiest to use the Poincare disc model, because then you can take two of the three edges of your triangle to lie in the x- and y-axes.  Or similarly in the UHP model you could let one edge lie in the y-axis and one edge lie in the unit circle.
