Elucid's fifth Postulate (Parallel Postulate): states the following: Given a line L and a point not on L, there is one and only one line L1 which contains the point, is in the same plane as L, and is parallel to L.

Nikolai Lobachevsky, on the contrary, stated the following:

• All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes - into cutting and non-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line.
• Lobachevsky's Parallel Postulate. There exist two lines parallel to a given line through a given point not on the line:

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Thus, both AE and AG are parallel to BC.

Lobachevsky's work was largely ignored until after his death, and provided Albert Einstein with a basis for his Theory of Relativity.