Euclid The famous mathematician Euclid is credited with being the first person to describe geometry.

Euclidean Geometry "flat" geometry

Parallel Postulate which states that through a given point not on a line, there is exactly one line passing through the point parallel to the given line.

Postulate: the sum of the angles of a triangle equals 180 o

What two Euclidean geometry postulates do spherical and hyperbolic geometry contradict? Explain how each of these postulates is contradicted by spherical and hyperbolic geometry. Write a conjecture about the sum of the measures of the angles of a triangle on a sphere.

Segments hyperbolic Explain the difference between a line segment in Euclidean geometry and in spherical geometry. In spherical geometry the shortest path between two points is not necessarily a unique path as in Euclidean geometry. When does this case occur?

Polygon A polygon in hyperbolic geometry is a sequence of points and geodesic segments joining those points. The geodesic segments are called the sides of the polygon. A triangle in hyperbolic geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and so on, as in Euclidean geometry. Here are some triangles in hyperbolic space: