Parallel postulate

If the sum of the interior angles α and β is less than 180°, 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;[1] it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23[2] just before the five postulates.[3]

Euclidean 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.[4]

This 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.[5]

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 180° (triangle postulate).
  3. There exists a triangle whose angles add up to 180°.
  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).[6][7]
  12. The Law of cosines, a general case of Pythagoras' Theorem.
  13. There is no upper limit to the area of a triangle. (Wallis axiom)[8]
  14. The summit angles of the Saccheri quadrilateral are 90°.
  15. 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)[9]

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 – constant separation, never meeting, same angles where crossed by some third line, or same angles where crossed by any third line – 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.