,
prove that the straight lines 
and 
meet at a certain point ." We consider a few of the many axioms equivalent to the original Postulate 5, the parallel lines axiom.
Euclid's axiom: 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.
You will note that these axioms sweep across most of plane geometry, revealing
the importance of this axiom to the geometry of triangles.
Playfair's Axiom: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
Playfair's Axiom was published by John Playfair in 1795 as an alternative to Euclid's parallel axiom. Playfair observed that he derived the axiom from Proclus. Proclus derived this version of the axiom in the fifth century. This version is also the same as Omar Khayyam's version from 1066. Through a point not on a line one and only one parallel can be drawn to this line.
Hilbert's Parallel Axiom: There can be drawn through any point A, lying outside of a line, one and only one line that does not intersect the given line.
In 1899, David Hilbert produced a set of axioms to characterize Euclidean geometry. His parallel axiom was one of these axioms.
Proclus' Axiom: If a line intersects one of two parallels it also intersects the other.
Proclus Diadochus (411-485) wrote Commentary on Euclid while teaching at Plato's
Academy. In his Commentary, he proves the following problem: "Given ,
prove that the straight lines
and
meet at a certain point ."
Clavius' axiom: All the points equidistant from a given straight line, on a given side of it constitute a straight line.
Christoph Clavius (1537-1612), of German birth, was the foremost mathematician of the Jesuit order. He wrote a number of textbooks, all of which went through numerous editions during his life. These include his version of Euclid's Elements. However, Clavius opposed the Copernican System on physical and scriptural grounds, and so he remained until near the end of his life.
John Wallis' axiom: Given a triangle we can construct a similar triangle of any size.
John Wallis proposed this axiom in the 1600's in order to prove the parallel axiom. Unfortunately, in his proof, he assumes Euclid's parallel axiom, which he is actually trying to prove, making the proof invalid.
Abu' Ali Ibn al-Haytham's axiom: Parallel straight lines are coplanar lines such that if produced indefinitely in both directions they do not intersect in either direction.
Around 1000 AD, Haytham, used line segments of finite magnitude to attempt to improve upon Euclid's parallel axiom. It was in pursuit of a proof for Euclid's parallel axiom, that the first theorems of hyperbolic geometry were derived.
Adrien-Marie Legendre (1752-1833) axiom: For any acute angle A and any point D in the interior of angle A, there exists a line through D and not through A which intersects both sides of angle A.
Adrien-Marie Legendre (1752-1833) made several attempts to prove the fifth postulate. The conclusion of the hypothesis is this: The sum of the angles of a triangle is equal to two right angles. This result shows as much as any how inextricably linked the parallel lines axiom is to the whole of classical plane geometry.
Clairaut's axiom: Rectangles Exist.
Alexis Claude Clairaut (1713-1765) was a French geometer who replaced the fifth postulate by his own postulate in 1741 in the text "Elements De Geometrie".
Farkas' axiom: Three non-collinear points always line on a circle.
(Bolyia Farkas 1775-1856).