[b]Theorem[/b]: In Hyperbolic Geometry the sum of the interior angles of a polygon is always less than [math]\left(p-2\right)\pi=\left(p-2\right)180^\circ[/math]. In particular, the sum of the interior angles of a hyperbolic triangle is always less than [math]\pi=180^\circ[/math].[br][br]No proof. Explore these examples. Adjust A, B and C, and observe that the sum of the angles always remains less than [math]180^\circ[/math].[br][br][b]Example 1[/b]:

[b]Example 2[/b]:

Questions:[br][br]What are the characteristics of triangles in the Poincaré Disk that lead to angle sums very close to 180 degrees?[br][br]What are the characteristics of triangles in the Poincaré Disk that lead to angle sums much smaller than 180 degrees?