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The 22 pages that link to Parallel postulate, each with the reason it gives.

• Non-Euclidean geometryRelated: Changing this axiom produces the principal non-Euclidean geometries.

• Hyperbolic geometryRelated: Rejecting its uniqueness produces the defining alternative of hyperbolic geometry.

• Spherical geometryRelated: On a sphere, great-circle lines intersect, so this postulate does not apply.

• Synthetic geometryRelated: Changing this assumption leads from Euclidean synthetic geometry to non-Euclidean geometries.

• Plane (geometry)Related: Its Euclidean form distinguishes the familiar plane from non-Euclidean geometries.

• Parallel linesRelated: It supplies the foundational rule behind uniqueness of parallels.

• Euclidean planeRelated: It distinguishes the Euclidean plane from non-Euclidean geometries.

• Elliptic geometryRelated: Elliptic geometry rejects this condition by allowing no parallel through the point.

• Nikolai LobachevskyRelated: Lobachevsky explored what follows when this postulate is replaced.

• Hyperbolic planeRelated: Replacing its uniqueness claim produces hyperbolic parallel behavior.

• Hilbert's axiomsRelated: Hilbert isolates the Euclidean parallel assumption as its own axiom group.

• János BolyaiRelated: Bolyai investigated what geometry follows when this postulate is replaced.

• Tarski's axiomsRelated: Its geometric content appears among the assumptions distinguishing the Euclidean system.

• Exterior angle theoremRelated: The standard proof relies on the Euclidean angle sum established using this postulate.

• Playfair's axiomRelated: Its equivalence shows that Playfair's wording expresses the same Euclidean assumption.

• Hjelmslev's theoremRelated: Replacing this postulate leads to the hyperbolic setting where the theorem holds.