Parallel postulate

In geometry, the parallel postulate is the fifth postulate in Euclid's Elements and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a straight line intersects two other straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. This may be also formulated as: If a straight line intersects two other straight lines, the two interior angles on the same side add to less than two right angles if and only if the two lines, if extended indefinitely, meet on that side. The difference between the two formulations lies in the converse of the first formulation: If a straight line intersects two other straight lines that intersect on some side of first line, the two interior angles on this side add to less than two right angles. This latter assertion is proved in Euclid's Elements by using the fact that two different lines have at most one intersection point. Conversely this latter assertion implies that two different lines cannot have two intersection points (draw a line passing between the two intersection points and apply the assertion to both sides of this line). This original formulation of the postulate does not specifically talk about parallel lines; however, its converse and the second formulation imply the existence of parallel lines, since, if the interior angles sum to two right angles, then the two lines do not intersect. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates. Euclidean geometry is a geometry that satisfies all of Euclid's axioms, including the parallel postulate and its converse. Non-Euclidean geometries are geometries that do not...