Comparison of the Geormetries

The sum of the measures of the angles of a triangle is 180o in Euclidean Geometry, less than 180o in hyperbolic geometry, and more than 180o in elliptic geometry. The area of triangle in hyperbolic geometry is proportional to the deficiency of its angle sum from 180 o ; the area of a triangle in elliptic geometry is proportional to the excess of its angle sum over 180 o . In Euclidean geometry all triangles have an angle sum of 180 o  irrespective of their area. Thus similar triangles with different areas can exist in Euclidean geometry; this occurence is not possible in either hyperbolic or elliptic geometry.

In two dimensional geometries, lines that are perpendicular to the same given are parallel in Euclidean geometry, are neither parllel nor intersecting in hyperbolic geometry, and intersect at the pole of the given line in elliptic gemetry. As indicated in the picture below, the appearance of the lines as straight or curved depends on the postulates for the space.

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