## Playfair's axiom |

in

,geometry **playfair's axiom**is an that can be used instead of the fifth postulate ofaxiom (theeuclid ):parallel postulate in a

, given a line and a point not on it, at most one lineplane to the given line can be drawn through the point.parallel ^{[1]}it is equivalent to euclid's parallel postulate in the context of

euclidean geometry ^{[2]}and was named after the scottishmathematician . the "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. the statement is often written with the phrase, "there is one and only one parallel". injohn playfair , two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.euclid's elements ^{[3]}^{[4]}this axiom is used not only in euclidean geometry but also in the broader study of

where the concept of parallelism is central. in the affine geometry setting, the stronger form of playfair's axiom (where "at most" is replaced by "one and only one") is needed since the axioms ofaffine geometry are not present to provide a proof of existence. playfair's version of the axiom has become so popular that it is often referred to asneutral geometry *euclid's parallel axiom*,^{[5]}even though it was not euclid's version of the axiom. a corollary of the axiom is that the of parallel lines is abinary relation .serial relation - history
- relation with euclid's fifth postulate
- transitivity of parallelism
- notes
- references

In **Playfair's axiom** is an

In a

plane , given a line and a point not on it, at most one lineparallel to the given line can be drawn through the point.^{[1]}

It is equivalent to Euclid's parallel postulate in the context of ^{[2]} and was named after the Scottish ^{[3]}^{[4]}

This axiom is used not only in Euclidean geometry but also in the broader study of *Euclid's parallel axiom*,^{[5]} even though it was not Euclid's version of the axiom.
A corollary of the axiom is that the