Polygon Angle Sums in Hyperbolic Geometry
Theorem: In Hyperbolic Geometry the sum of the interior angles of a polygon is always less than . In particular, the sum of the interior angles of a hyperbolic triangle is always less than .
No proof. Explore these examples. Adjust A, B and C, and observe that the sum of the angles always remains less than .
Example 1:
Example 2:
Questions:
What are the characteristics of triangles in the Poincaré Disk that lead to angle sums very close to 180 degrees?
What are the characteristics of triangles in the Poincaré Disk that lead to angle sums much smaller than 180 degrees?