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Euclid's parallel axiom is false in non-Euclidean geometry because non-Euclidean geometry occurs within a different theory of space. There may be one absolute occurrence in non-Euclidean space where Euclid's parallel axiom is valid. Possibly as some form of infinity.

Q: Why is the parallel axiom in Euclid's geometry false in non-Euclidian geometry?

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Yes they are. It is delineated in something called the parallel postulate, and the axiom is also called Euclid's fifth postulate. This is boilerplate Euclidean geometry, and a link can be found below if you'd like to review the particulars.

The Playfair Axiom (or "Parallel Postulate")

it is called an axiom

Probably the best known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states: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.

axiom

Related questions

The Euclidean Parallel Axiom states that through a point not on a given line, there exists exactly one line parallel to the given line. This axiom is one of the five postulates in Euclidean geometry that forms the foundation for the study of parallel lines and geometry.

An axiom.

Not in Euclidean Geometry. Euclid's 5th axiom is that parallel lines never meet. However, unlike the first 4 axiom, it is impossible to prove the 5th axiom; depending upon the situation, you can either assume that parallel lines meet or don't; when they do meet, there are some very interesting consequences (for example, the possibility of a hyperbolic space). To my knowledge, if they meet, they are intersecting/perpendicular lines.

Yes they are. It is delineated in something called the parallel postulate, and the axiom is also called Euclid's fifth postulate. This is boilerplate Euclidean geometry, and a link can be found below if you'd like to review the particulars.

An axiom of Euclidean geometry.

"A geometry rule that we accept to be true without any proof is called an axiom." "The axiom 'might makes right' is often true in the governments of the world.

Playfair Axiom

parallel postulate

Another name for the Playfair Axiom is the Euclid's Parallel Postulate. It states that given a line and a point not on that line, there is exactly one line parallel to the given line passing through the given point.

It is an axiom that parallel lines never meet in Euclidean geometry. Never.However in another kind of geometry (can't remember name) it states that parallel lines will eventually meet.Take a look at this picture in the related link, below.Technically the lines are parallel (in theory, they have imperfections), but due to our perspective parallel lines appear to meet. Note: If they really do meet, then you could drive down the road and eventually there would not be a road, anymore.

An axiom, in Geometry, is a statement that we assume is true. Whether it is actually true or not is irrelevant. For the purpse of solving the problem, it is considered to be true.

The Playfair Axiom (or "Parallel Postulate")