Euclide geometry elements

The sum of the angles of a triangle is 180 degrees, right? Not in the world of non-Euclidean geometry. Non-Euclidean geometry is based on Euclidean geometry, the geometry most of us learn in high school. Before you can understand non-Euclidean geometry, you must first understand the five basic postulates of Euclidean geometry:
1) You can draw a straight line between any two points
2) You can extend any straight line segment infinitely
3) You can draw a circle using any straight line segment as the radius and one of the endpoints as the center
4) All right angles are congruent (equivalent)
5) Given a line and and a point not on the line, you can draw one and only one line parallel to the line through the point
Non-Euclidean geometry accepts the first four of Euclid's postulates but not the fifth. Instead, one of two alternate postulates is used to create a cohesive and consistent alternate geometry:
1) Given a line and a point not on the line, you cannot draw any lines parallel to the line through the point
2) Given a line and a point not on the line, you can draw two or more lines parallel to the line through the point
The first alternate postulate leads to an alternate geometry called spherical geometry (because it can be used to easily describe the surface of a sphere). Spherical geometry leads to a system where the sum of the angles of a triangle is more than 180 degrees. Spherical geometry is frequently used to describe the surface of the Earth.
The second alternate postulate leads to an alternate geometry called hyperbolic geometry. Hyperbolic geometry leads to a system where the sum of the angles of a triangle is less than 180 degrees. Hyperbolic geometry doesn't map easily to a physical system, but many cosmologists believe that the geometry of the universe can be best described by hyperbolic geometry.
Because most of us are only familiar with Euclidean geometry (also called flat geometry), these alternate geometries where the sum of the angles of a triangle do not add up to 180 degrees are complete and consistent mathematical descriptions of the world.