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thumb|right|350px|If the sum of the interior angles Î± and Î² is less than 180Â°, the two straight lines, produced indefinitely, meet on that side.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 in Euclidean geometry. It states that, in two-dimensional geometry:If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.This postulate does not specifically talk about parallel lines;weblink" title="web.archive.org/web/20170202075326weblink">non-Euclidean geometries, by Dr. Katrina Piatek-Jimenez it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23Euclid's Elements, Book I, Definition 23 just before the five postulates.Euclid's Elements, Book IEuclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or, in other places known as neutral geometry).

## Equivalent properties

Probably the best known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.Euclid's Parallel Postulate and Playfair's AxiomThis axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the remaining axioms which give Euclidean geometry, each of these can be used to prove the other, so they are equivalent in the context of absolute geometry.{{harvnb|Henderson|TaimiÅ†a|2005|loc=pg. 139}}Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. These equivalent statements include:
1. There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)
2. The sum of the angles in every triangle is 180Â° (triangle postulate).
3. There exists a triangle whose angles add up to 180Â°.
4. The sum of the angles is the same for every triangle.
5. There exists a pair of similar, but not congruent, triangles.
6. Every triangle can be circumscribed.
7. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
8. There exists a quadrilateral in which all angles are right angles, that is, a rectangle.
9. There exists a pair of straight lines that are at constant distance from each other.
10. Two lines that are parallel to the same line are also parallel to each other.
11. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).{{Citation

| title = CRC concise encyclopedia of mathematics
| author = Eric W. Weisstein
| page = 2147
| quote = The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem.
| edition = 2nd
| isbn = 1-58488-347-2
| year = 2003
}}{{Citation
| title = The principle of sufficient reason: a reassessment
| author = Alexander R. Pruss
| quote = We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate.
| isbn = 0-521-85959-X
| year = 2006
| publisher = Cambridge University Press
| page = 11
}}
1. The Law of cosines, a general case of Pythagoras' Theorem.
2. There is no upper limit to the area of a triangle. (Wallis axiom)WEB

, Euclid's Fifth Postulate
, 30 September 2011
, Alexander Bogomolny, Bogomolny, Alexander
, Cut The Knot
,
1. The summit angles of the Saccheri quadrilateral are 90Â°.
2. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus' axiom)WEB

, Proclus' Axiom â€“ MathWorld
, 2009-09-05
, Weisstein
, Eric W.
, However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the four common definitions of "parallel" is meant â€“ constant separation, never meeting, same angles where crossed by some third line, or same angles where crossed by any third line â€“ since the equivalence of these four is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate. In the list above, it is always taken to refer to non-intersecting lines. For example, if the word "parallel" in Playfair's axiom is taken to mean 'constant separation' or 'same angles where crossed by any third line', then it is no longer equivalent to Euclid's fifth postulate, and is provable from the first four (the axiom says 'There is at most one line...', which is consistent with there being no such lines). However, if the definition is taken so that parallel lines are lines that do not intersect, or that have some line intersecting them in the same angles, Playfair's axiom is contextually equivalent to Euclid's fifth postulate and is thus logically independent of the first four postulates. Note that the latter two definitions are not equivalent, because in hyperbolic geometry the second definition holds only for ultraparallel lines.

## History

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- Euclidian and non euclidian geometry.png -
Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.

## Converse of Euclid's parallel postulate

thumb|right|350px|The converse of the parallel postulate: If the sum of the two interior angles equals 180Â°, then the lines are parallel and will never intersect.Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De MorganHeath, T.L., The thirteen books of Euclid's Elements, Vol.1, Dover, 1956, pg.309. pointed out, this is logically equivalent to (Book I, Proposition 16). These results do not depend upon the fifth postulate, but they do require the second postulateCoxeter, H.S.M., Non-Euclidean Geometry, 6th Ed., MAA 1998, pg.3 which is violated in elliptic geometry.

## Criticism

Attempts to logically prove the parallel postulate, rather than the eighth axiom,Schopenhauer is referring to Euclid's Common Notion 4: Figures coinciding with one another are equal to one another. were criticized by Arthur Schopenhauer. However, the argument used by Schopenhauer was that the postulate is evident by perception, not that it was not a logical consequence of the other axioms.

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## References

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|last=Smith
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}}

{{Citation
|last=Eder
|first=Michelle
|year=2000
|title=Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam
|url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html
|publisher=Rutgers University
|accessdate=2008-01-23
}}

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