
The parallel postulate is the fifth postulate of Euclidean geometry. It states, That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Euclid. Elements, Book I.^{[1]}Stated in simpler language, if the sum of the interior angles on the same side of a transversal of two lines is less than 180°, the two lines meet on that side.^{[2][4]} If the sum of the angles is equal to 180°, the two lines do not meet, and so are parallel. If the sum of the angles is greater than 180°, the two lines meet on the opposite side. In modern geometry, this postulate is called the axiom of parallels and is stated differently: In plane α there can be drawn through any point A, lying outside of a straight line a, one and only one line that does not intersect line a. This line is called parallel to a through given point A.^{[3]} 
Diagram  Sum of interior angles  Lines meet ... 

α+β < 180°  Lines meet on the same side as the interior angles.  
α+β = 180°  Lines do not meet. They are parallel.  
α+β > 180°  Lines meet on the opposite side of the interior angles.  
Table 1: Cases of the Parallel Postulate. 
There are a number of geometric properties that are equivalences of the parallel postulate. Two properties are equivalent if one implies the other. Some of the equivalencies of the parallel postulate are:
The parallel postulate has been shown to be very important in the definitions of geometries. Because it is not intuitively obvious like the first four postulates, many mathematicians believed that the parallel postulate could be proved using the first four postulates. There were many attempts at this proof that were unsuccessful.
Starting in 1829, mathematicians switched from trying to prove the fifth postulate to exploring geometries that do not contain the parallel postulate. As a result, two valid geometries were discovered: hyperbolic geometry and elliptical geometry.
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