Playfair's axiom
E519271
Playfair's axiom is a reformulation of Euclid’s parallel postulate stating that through a point not on a given line there is exactly one line parallel to the given line, fundamental to Euclidean geometry.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Euclid's fifth postulate | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
geometric axiom
ⓘ
parallel postulate formulation ⓘ |
| assumes | existence of at least one line through any two distinct points ⓘ |
| category | axiom of plane geometry ⓘ |
| clarityAdvantage | more intuitive and concise than Euclid's original fifth postulate ⓘ |
| consequence |
circles of equal radius are congruent in Euclidean geometry
ⓘ
rectangles exist in Euclidean geometry ⓘ |
| contradictedBy |
axioms of elliptic geometry
ⓘ
axioms of hyperbolic geometry ⓘ |
| domainOfDiscourse | points and lines in a Euclidean plane ⓘ |
| equivalentTo | various other Euclidean parallel axioms ⓘ |
| expresses | uniqueness of parallels through an external point ⓘ |
| field | Euclidean geometry NERFINISHED ⓘ |
| formalization | can be expressed in first-order logic over incidence and parallelism relations ⓘ |
| geometricConsequence |
angle sum of polygons depends only on number of sides in Euclidean geometry
ⓘ
area of a triangle determined by its base and height in Euclidean geometry ⓘ existence of similar figures of arbitrary size ⓘ |
| historicalPeriod | 18th century formulation ⓘ |
| implies |
Pythagorean theorem in Euclidean geometry
ⓘ
existence of similar but non-congruent triangles in Euclidean geometry ⓘ sum of angles in a triangle equals 180 degrees in Euclidean geometry ⓘ |
| logicalForm | uniqueness axiom for parallels ⓘ |
| logicalStatus | equivalent to Euclid's parallel postulate in Euclidean geometry ⓘ |
| logicalType | independent of Euclid's first four postulates ⓘ |
| namedAfter | John Playfair NERFINISHED ⓘ |
| parallelLineCountInEllipticGeometry | no parallels through a point not on a given line ⓘ |
| parallelLineCountInHyperbolicGeometry | infinitely many parallels through a point not on a given line ⓘ |
| quantifierProperty | asserts existence and uniqueness of a parallel line ⓘ |
| reformulationOf | Euclid's parallel postulate NERFINISHED ⓘ |
| relatedConcept |
Euclid's Elements
NERFINISHED
ⓘ
elliptic geometry ⓘ hyperbolic geometry ⓘ non-Euclidean geometry ⓘ parallel lines ⓘ |
| requires |
notion of incidence between points and lines
ⓘ
notion of parallel lines ⓘ |
| roleInTheory | distinguishes Euclidean geometry from non-Euclidean geometries ⓘ |
| states | Through a point not on a given line there is exactly one line parallel to the given line ⓘ |
| teachingUse | standard form of the parallel postulate in modern geometry education ⓘ |
| truthValueInEllipticGeometry | false ⓘ |
| truthValueInEuclideanGeometry | true ⓘ |
| truthValueInHyperbolicGeometry | false ⓘ |
| usedAs | modern replacement for Euclid's fifth postulate in many textbooks ⓘ |
| usedFor | characterizing flat (zero curvature) geometric spaces ⓘ |
| usedIn | axiomatic treatments of plane geometry ⓘ |
| usedWith | Euclid's first four postulates ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Euclid's fifth postulate