What is Parallel postulate?
[[Image:Parallel postulate en.svg|thumb|right|350px|If the sum of the inner angles α (''alpha'') and β (beta) is less than 180°, the two lines will intersect somewhere, if both are prolonged to [[infinity]].]] In [[geometry]] the '''parallel postulate''' is one of the [[axiom]]s of [[Euclidean geometry]]. Sometimes it is also called '''[[Euclid]]'s fifth postulate''', because it is the fifth postulate in Euclid's [[Euclid's Elements|Elements]]. The postulate says that: ''If you cut a line segment with two lines, and the two interior angles the lines form add up to less than 180°, then the two lines will eventually meet if you extend them long enough.'' The field of geometry which follows all of Euclid's axioms is called [[Euclidean geometry]]. Geometries which do not follow all of Euclid's axioms are called [[non-Euclidean geometry]]. ==History== Some mathematicians thought that Euclid's fifth postulate was much longer and more complicated than the other four postulates. Many of them thought that it could be proven from the other simpler axioms. Those who tried to do it included [[Omar Khayyám]], and later [[Giovanni Gerolamo Saccheri]], [[John Wallis]], [[Lambert]], and Legendre. Some mathematicians announced that they had proven the proposition from the simpler propositions, but they all turned out to be mistaken. ===Playfair's axiom===<!-- [[Playfair's axiom]] redirects here --> Another more recent proposition known as ''Playfair's axiom'' is similar to Euclid's fifth postulate. It says that: ''Given a straight line and a point not on this line, you can only draw one straight line through this point that will not meet the other straight line.'' In fact, mathematicians found out that this axiom is not only similar to Euclid's fifth postulate, but has exactly the same implications. Mathematically, the two propositions are called 'equivalent' propositions. Today Playfair's axiom is used more often by mathematicians than Euclid's original parallel postulate. ===Non-Euclidean geometry=== Eventually s