a and b are parallel, the transversal t produces congruent angles.

In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. It states that: Two parallel lines cut by a transversal, with plenty of labels. ... Two parallel lines cut by a transversal, with plenty of labels. ... Transversal t cuts two parallel lines, a and b. ... Table of Geometry, from the 1728 Cyclopaedia. ... Euclid Euclid of Alexandria (Greek: ) (ca. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... Euclid Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. ...

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. In mathematics, a line segment is a part of a line that is bounded by two end points. ... A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ... This article is about angles in geometry. ...

Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate cannot hold is known as a non-euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the first four postulates) is known as absolute geometry (or, in some places, neutral geometry). Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... Absolute geometry is a geometry that doesnt assume the parallel postulate. ... Absolute geometry is a geometry that doesnt assume the parallel postulate. ...

Converse of Euclid's parallel postulate

Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. A proof appears in the Elements of an equivalent statement: Any two angles of a triangle are together less than two right angles. The proof depends on an earlier proposition: In a triangle ABC, the exterior angle at C is greater than either of the interior angles A or B. This in turn depends on Euclid's unstated assumption that two straight lines meet in only one point, a statement not true of elliptic geometry. In traditional logic Conversion is a form of immediate inference in which from a given categorical proposition another proposition is inferred which has as its subject the predicate of the original proposition, and has as its predicate the subject of the original proposition, with the quality of the proposition remaining... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... This article is about angles in geometry. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... This article is about angles in geometry. ... In geometry, an internal angle is an angle that 2 sides of a polygon form by touching. ...

Logically equivalent properties

Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proven in a system where the parallel postulate is true, and that if they are assumed as axioms, then the parallel postulate can be proven. There are, in addition, properties that are equivalent to the conjunction of Euclid's parallel postulate and its converse, and thus can be used to distinguish Euclidean geometry from both elliptic geometry and hyperbolic geometry simultaneously. One of the most important of these properties, and the one that is most often assumed today as an axiom, is Playfair's axiom, named after the Scottish mathematician John Playfair. It states: In logic, statements p and q are logically equivalent if they have the same logical content. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... Professor John Playfair FRSE (March 10, 1748 – July 20, 1819) was a Scottish scientist. ...

Exactly one line can be drawn through any point not on a given line parallel to the given line. Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ...

Some of the other statements that are equivalent to the parallel postulate appear at first to be unrelated to parallelism. Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. Here are some of these results: In epistemology, a self-evident proposition is one that can be understood only by one who knows that it is true. ... The unconscious mind (or subconscious) is the aspect (or puported aspect) of the mind of which we are not directly conscious or aware. ...

1. The sum of the angles in every triangle is 180°.
2. There exists a triangle whose angles add up to 180°.
3. The sum of the angles is the same for every triangle.
4. There exists a pair of similar, but not congruent, triangles.
5. Every triangle can be circumscribed.
6. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
7. There exists a quadrilateral of which all angles are right angles.
8. There exists a pair of straight lines that are at constant distance from each other.
9. Two lines that are parallel to the same line are also parallel to each other.
10. Given two parallel lines, any line that intersects one of them also intersects the 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).
12. There is no upper limit to the area of a triangle. [1]

It should be noted, however, that the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the three common definitions of "parallel" is meant - constant separation, never meeting or same angles where crossed by a third line - since the equivalence of these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate! This article is about angles in geometry. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... Several equivalence relations in mathematics are called similarity. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... A black circle circumscribed by a red square In geometry, a circumscribed planar shape or solid is one that encloses and fits snugly around another geometric shape or solid. ... In geometry, a quadrilateral is a polygon with four sides and four vertices. ... This article is about angles in geometry. ... The distance between two points is the length of a straight line segment between them. ... Right-angled triangle A triangle in Euclidean geometry where the measure of one of the angles is 90º. This type of triangle is very popular in highschool level mathematics when the Pythagorean Theorem is introduced. ... There are thousands of proofs of the Pythagorean theorem. ... Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ...

History

For two thousand years the parallel postulate was suspected by some mathematicians to be a theorem which could be proved using Euclid's first four postulates. A great many attempts were made to provide such a proof, constituting one of the largest collections of writings on any single topic in mathematics. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...

The main reason such a proof was so highly sought after was that while Euclid's other postulates appeared self-evident and intuitively obvious, the fifth postulate essentially described the intersection of lines at potentially infinite distances, a concept that could hardly be called self-evident. In addition, the converse of the fifth postulate is a theorem that was proved by Euclid in Book I of the Elements (Proposition 17).

Archimedes, in his treatise On Parallel Lines, defined parallel lines as those equidistant to each other everywhere. From this the parallel postulate can be "proved" if you are willing to accept that a "line" equidistant to a straight line everywhere is in fact a straight line. Archimedes (Greek: Αρχιμήδης ) (c. ...

Omar Khayyám (1050-1123) recognized that three possibilities arose from omitting Euclid's Fifth; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse. He persuaded himself that the acute and obtuse cases lead to contradiction, but had made a tacit assumption equivalent to the fifth to get there. Girolamo Saccheri (1667-1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (because, with Euclid, he'd unwittingly assumed lines can be extended indefinitely and have infinite length); but failed to debunk the acute case (but managed to wrongly persuade himself that he had). InsertformulahereInsertformulahere Tomb of Omar Khayyám, Nishapur, Iran. ... Giovanni Gerolamo Saccheri (September 5, 1667 – October 25, 1733) was an Italian Jesuit priest and mathematician. ...

Where Saccheri and Khayyám had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries which result. In 1829 Nikolai Ivanovich Lobachevski published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German). In 1831 János Bolyai included, in a book by his father, an appendix describing acute geometry, which he had doubtless developed independently of Lobachevski. It is probable that Carl Friedrich Gauss had actually studied the problem before that, but he didn't publish. The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and spherical geometry (the obtuse case). Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (December 1, 1792–February 24, 1856 (N.S.); November 20, 1792–February 12, 1856 (O.S.)) was a Russian mathematician. ... János Bolyai (December 15, 1802–January 27, 1860) was a Hungarian mathematician. ... (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (December 1, 1792–February 24, 1856 (N.S.); November 20, 1792–February 12, 1856 (O.S.)) was a Russian mathematician. ... Bernhard Riemann. ... Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ...

The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868. For more information, see the history of non-Euclidean geometry. Eugenio Beltrami (16 November 1835 - 18 February 1900) was an Italian mathematician notable for his work on non-Euclidean geometry, electricity, and magnetism. ... 1868 (MDCCCLXVIII) was a leap year starting on Wednesday (see link for calendar) of the Gregorian calendar or a leap year starting on Friday of the 12-day-slower Julian calendar. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...