Parallel postulate

If a line segment intersects two straight lines forming two interior angles on the same side sum to less than two right angles then the two lines segments, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

The parallel postulate is the only postulate of Euclidean geometry which fails for non-Euclidean geometry.

Several properties of Euclidean geometry are logically equivalent to Euclid's parallel postulate, meaning that they can be proved in a system where the parallel postulate is true, and that if they are assumed as axioms, then the parallel postulate can be proved. 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:

A line can be drawn through any point not on a given line parallel to the given line.

Many attempts were made to prove the parallel postulate in terms of Euclid's first four postulates. The independent discovery of non-Euclidean spaces by Gauss and Lobachevsky finally demonstrated the independence of the parallel postulate. (See "History" under non-Euclidean geometry for further discussion.)

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 proved the parallel postulate from Euclid's other postulates. Here are some of these results:

1. The sum of the angles in a triangle is 180°.
2. The sum of the angles is the same for every triangle.
3. There exists a pair of similar, but not congruent, triangles.
4. Every triangle can be circumscribed[?].
5. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
6. There exists a pair of straight lines that are at constant distance from each other.