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: 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. ... Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... Euclid of Alexandria (Greek: ) (ca. ... Euclids Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

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 are the angles less than the 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, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... 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 is violated 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 affine 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. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ...

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. 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. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... A mathematician is a person whose area of study and research is mathematics. ... John Playfair (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. The term Parallel has a number of important meanings: Parallel (geometry) occurs in geometry. ...

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 a 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).

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 two-dimensional figure 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. ... Other uses: Quadrilateral (disambiguation) In geometry, a quadrilateral is a polygon with four sides and four vertices. ... This article is about angles in geometry. ... personal space, proxemics. ... 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. ...

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).

Omar Khayyam (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).

Where Saccheri and Khayyam had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw folk 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. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist 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, 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 was a leap year starting on Wednesday (see link for 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. ...