---
tag: 2PC3XY0PYZ
kind: statement
role: fact
name: Parallel postulate
msc: [51M05, 51F05, 01A20]
uses: [0MP9BCF6DP, 0GJAJC6TNZ, PR7Q3QMEHZ, VKFPCSRNDH]
about: [KRZD96GSVC]
sources:
  - url: https://en.wikipedia.org/wiki/Parallel_postulate
same_as:
  wikidata: Q207014
  wikipedia: Parallel postulate
---

## Statement

Let two [[lines|0MP9BCF6DP]] in a plane be cut by a third line, the transversal.
Suppose that the two interior [[angles|0GJAJC6TNZ]] on one side of the transversal have a sum less than two right angles.
Then the two lines, extended far enough, meet on that side of the transversal.

## Remarks

This is the fifth postulate of Book I of [[Euclid's Elements|83WHDAV68J]], an axiom and not a theorem.
Over the axioms of [[absolute geometry|VKFPCSRNDH]], it is equivalent to [[Playfair's axiom|PR7Q3QMEHZ]], 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|GN4N94GQ0Z]] 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.

## Explanations

### L1 human

Take two lines and a third line that crosses both.
If the two lines lean toward each other on one side, they meet on that side.
This sounds obvious, but it cannot be proved from Euclid's other rules.

### L2 human

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.