Hyperbolic General Tiangle Measured
Properties of All Triangles in Hyperbolic Geometry
Above is a general triangle in the Half-plane model of Hyperbolic Geometry. Line segments are arcs of circles centered on the Hyperbolic Half-plane Edge (x-axis) or they are vertical Euclidean line segments. The measure of an angle is given by the measure of the angle between Euclidean tangent rays.
What properties are shared by all Hyperbolic triangles?
Is the longest side always opposite the largest angle and the shortest side opposite the smallest angle?
Investigate the sum of the measures of the interior angles. Is it pi = 180 degrees? Is it constant? If not, what range of values can the sum take on? Can you figure out what is meant by "defect"?