## Parallel postulate |

In **parallel postulate**, also called ** Euclid's fifth postulate** because it is the fifth postulate in

If aline segment intersects two straightlines forming two interior angles on the same side that sum to less than tworight 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

- equivalent properties
- history
- converse of euclid's parallel postulate
- criticism
- see also
- notes
- references
- external links

Probably the best known equivalent of Euclid's parallel postulate, contingent on his other postulates, is

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 ^{[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

- There is at most one line that can be drawn parallel to another given one through an external point. (
Playfair's axiom ) - The sum of the
angles in everytriangle is 180° (triangle postulate ). - There exists a triangle whose angles add up to 180°.
- The sum of the angles is the same for every triangle.
- There exists a pair of
similar , but notcongruent , triangles. - Every triangle can be
circumscribed . - If three angles of a
quadrilateral areright angles , then the fourth angle is also a right angle. - There exists a quadrilateral in which all angles are right angles, that is, a
rectangle . - There exists a pair of straight lines that are at constant
distance from each other. - Two lines that are parallel to the same line are also parallel to each other.
- 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]} - The
Law of cosines , a general case of Pythagoras' Theorem. - There is no upper limit to the
area of a triangle. (Wallis axiom )^{[8]} - The summit angles of the
Saccheri quadrilateral are 90°. - 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