---
tag: VKFPCSRNDH
kind: concept
name: Absolute geometry
msc: [51F05, 51F10]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_geometry
same_as:
  wikidata: Q332430
  wikipedia: Absolute geometry
---

## Summary

Absolute geometry, also called neutral geometry, is the geometry that [[Hilbert's axioms|KM1FB8XZ2R]] give without the axiom of parallels.
Its theorems hold both in [[Euclidean geometry|KRZD96GSVC]] and in [[hyperbolic geometry|GN4N94GQ0Z]].
It contains the congruence criteria for triangles, the [[exterior angle theorem|0NTSE7EREA]] and the [[Saccheri–Legendre theorem|1Y2XK0QJDX]], but not the angle sum $180^\circ$.

## Explanations

### L1 human

Absolute geometry is what remains of school geometry when one forgets the rule about parallel lines.
Everything proved there is true both on a flat plane and in the curved world of hyperbolic geometry.
For example, in any triangle an exterior angle is larger than each of the two opposite interior angles.
On the other hand, "the angles of a triangle add up to $180^\circ$" is not a theorem of absolute geometry.