Geometry are, under some circumstances, invalid in Hyperbolic Geometry.As described within the preceding text, there are many different variations in Geometry. The term “Geometry” is very broad and can be quite speculated into other means. Mainly, there are two breeds of Geometry. The standardized Euclidean Geometry, which is mostly accepted and used by all, and there is the Non-Euclidean Geometry, anything other than the standardized Euclidean methods. Non-Euclidean geometry are not often studied and utilized as much as the Euclidean means of calculations. Most High School courses only study the areas of Euclidean Geometry and don’t get into the sophisticated area of any Non-Euclidean Geometry. The Non-Euclidean Geometry mainly focused on in the preceding text was that of Spherical and Hyperbolic. There are many other Non-Euclidean Geometries which can be studied, some being Demonstrative, Analytic, Descriptive and Conic, to name a few. Because of their extensive comprehending language, most people try to isolate themselves from studying any means of Non-Euclidean Geometry to avoid confusion between that and Euclidean Geometry.Appendix 1-1:The sphere on the left is a perfect example of a triangle containing angles which sum is greater than 180.Appendix 1-2:The picture on the left demonstrates how in Spherical geometry, many things occur. In the uninteresting case the plane and the sphere miss each other. If they do meet each other there are two possibilities. First they can meet in a single point. In this case the plane is tangent to the sphere at the point of intersection. In the other case the sphere and the plane meet in a circle. Appendix 1-3:If A and B are two points on the sphere, then the distance between them is the distance along the great circle connecting them. Since this circle lies totally in a plane, we can figure this distance using the plane figure to our left. If the angle ACB is a, and if a i...
