In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom...

evident in his criticism of contemporaneous attempts to prove the parallel postulate in Euclidean geometry. Writing shortly before the discovery of hyperbolic...

seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus...

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments...

instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line and a point not on it, at most one line parallel to the given line...

direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine...

triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. In the...

seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute...

affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former...

without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was...

consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent...

geometry sufficient to resolve the ancient problem of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry...

and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding...

parallel postulate from Euclid's Elements is equivalent to the statement that given a straight line and a point not on that line, only one parallel to...

or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point...

assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different...

ISBN 1-58488-347-2. The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean...

Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any...

the Parallel Postulate, as shown by Max Dehn. In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate. As...

comparison of magnitudes. While postulates 1 through 4 are relatively straightforward, the 5th is known as the parallel postulate and particularly famous. Book...

particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect...

that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass through the point...

vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum. Such a quadrilateral is sometimes...

constructibility Rank-into-rank Kripke–Platek axioms Diamond principle Parallel postulate Birkhoff's axioms (4 axioms) Hilbert's axioms (20 axioms) Tarski's...

a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must...

impossible to prove (parallel postulate). After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized...

Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative...

geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact...

Convergence of parallel lines can refer to: In everyday life, the vanishing point phenomenon Non-Euclidean geometry in which Euclid's parallel postulate does not...

no need to break into different cases, such as parallel to the x {\displaystyle x} -axis and non-parallel. The other advantage is that these homogeneous...