Thread: There are always parallels, even without the parallel postulate...right?

A while ago I heard about elliptic geometry, a geometry in which there are no parallels. This is not possible, even without a parallel postulate, right? That is, aren't there always parallels, even in absolute geometry?

This is my reasoning: consider lines AB and q, intersecting at point E, which is between A and B. Then choose point K somewhere else on line r, and construct points L and M, such that L and A are on one side of r, M and B are on the other, and the angle AEK is of the same measure as its alternate interior angle MKE.

If there are no parallels, let lines AB and LM intersect at a point X, forming triangle EKX with the exterior angle at E of the same measure as the remote interior angle at K. This contradicts proposition 16 of book 1 of the Elements (link), which is established without the use of any parallel postulate.

So my question is this: how does this elliptic geometry deal with this contradiction? Does it omit another of the postulates of Euclidean geometry (for a total of two)?

Originally Posted by benzi455

A while ago I heard about elliptic geometry, a geometry in which there are no parallels. This is not possible, even without a parallel postulate, right? That is, aren't there always parallels, even in absolute geometry?

This is my reasoning: consider lines AB and q, intersecting at point E, which is between A and B. Then choose point K somewhere else on line r, and construct points L and M, such that L and A are on one side of r, M and B are on the other, and the angle AEK is of the same measure as its alternate interior angle MKE.

If there are no parallels, let lines AB and LM intersect at a point X, forming triangle EKX with the exterior angle at E of the same measure as the remote interior angle at K. This contradicts proposition 16 of book 1 of the Elements (link), which is established without the use of any parallel postulate.

So my question is this: how does this elliptic geometry deal with this contradiction? Does it omit another of the postulates of Euclidean geometry (for a total of two)?

See the second section of this.

The gist of what it says is:

Converse of Euclid's parallel postulate

Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement: Any two angles of a triangle are together less than two right angles. The proof depends on an earlier proposition: In a triangle ABC, the exterior angle at C is greater than either of the interior angles A or B. This in turn depends on Euclid's unstated assumption that two straight lines meet in only one point, a statement not true of elliptic geometry.

RonL

Thanks. I actually noticed that that lines intersect at more than one point in elliptic geometry, but I never thought to put this together with the absence of parallel lines. This was probably because, although there are supposedly only five postulates necessary in Euclidean geometry, SAS triangle congruence (or the idea that all angles and line segments of equal measure are congruent and vice versa) and others are also needed, including the idea that lines may never have at least one point at which they don't intersect, but more than one point at which they do. Thanks for the quick reply!

--Benzi