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)?