Parallel postulate

Remarks

This is the fifth postulate of Book I of Euclid's Elements, an axiom and not a theorem. Over the axioms of absolute geometry, it is equivalent to Playfair's axiom, and also to the statement that the angle sum of every triangle is two right angles. It is independent of the other axioms: the Euclidean plane satisfies it, and the hyperbolic plane satisfies all axioms of absolute geometry but not this one. Many attempts to prove it from the other postulates, such as those of Saccheri and Lambert, led in the end to the discovery of non-Euclidean geometry.

Going deeper

For a student

The postulate is the dividing line between Euclidean and hyperbolic geometry. Results such as the angle sum $180^\circ$ of a triangle, the existence of rectangles and the theory of similar triangles depend on it. Results of absolute geometry, such as the congruence criteria for triangles and the exterior angle theorem, do not. In proofs one usually uses the equivalent Playfair's axiom, which is easier to apply.