Equivalents to Euclid’s Postulate V
Under the assumption that Euclid’s first four postulates hold, each of the following is
equivalent to Euclid’s Postulate V:
1) Playfair’s axiom: given a line l and a point P not on l, there is exactly one line
through P parallel to l. (Note: parallel means never intersecting).
2) The set of points equidistant from a given line forms a line.
3) The sum of the interior angles of a triangle is 180 degrees (two right angles).
4) Wallis’s postulate on the existence of similar triangles: given any triangle, there
exists a similar triangle of arbitrary magnitude. (Here magnitude can be taken to
mean the length of one of the sides.)
5) The converse of the alternate interior angles theorem (Euclid’s Proposition 29 –
see handout on Euclid’s Elements).
6) The perpendicular bisectors of the three sides of a triangle are concurrent (i.e.
have a common point of intersection).
7) Given any three noncollinear points, there exists a point equidistant from these
three points.
8) If point C is on a circle with diameter AB and if C does not lie on AB, then
ACB is a right angle.
9) There exists a rectangle (a quadrilateral with four right angles).
10) Any pair of parallel lines has a common perpendicular.
11) The Pythagorean theorem.
12) Whenever a line is perpendicular to one of two parallel lines, it must be
perpendicular to the other. (Exercise 2.1.7)
13) Two lines that are parallel to the same line are either coincident or are themselves
parallel. (Exercise 2.1.9)
14) If a line intersects but is not coincident with one of two parallel lines, it must
intersect the other. (Exercise 2.1.10)
Reference: John McCleary, Geometry from a Differentiable Viewpoint, Cambridge
University Press 1994.