Hyperbolic Geometry and Proposition 29
Hyperbolic Geometry Models
-Saddle
-Poincare Disc
-Klein Model
-Pseudosphere
Proposition 29
A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.
Pseudosphere
Proposition 29 Failing
Proposition 29 is the first proposition to use postulate 5. This proposition relies on there being exactly one parallel line parallel to another that goes through a given point. However in hyperbolic geometry, there can be more than one parallel line through a given point. This means that alternate angles aren't always equal between parallel lines as they are in Euclidean geometry. Therefore, proposition 29 fails in hyperbolic geometry.