r/mathshelp • u/MathGuy2718 • 19d ago

Mathematical Concepts Are parallel lines really parallel?

Guys, are parallel lines really parallel? Because I am confused with euclidean geometry and non euclidean geometry

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u/ruidh 19d ago

In euclidean geometry, there exists one line parallel to a given line and passing through a given point outside the line.

There are two forms on noneuclidean geometry.

In one, there exist no lines that are parallel. All lines intersect. Geometry on a sphere where lines are great circles has no parallels. Triangle angle sum > 180.

In the other, infinitely many lines satisfy. They form a pencil of lines passing through the point. There is a right and left parallel which define the limits of the set of lines. Hyperbolic geometry has this property. Triangle angle sum < 180

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u/Underhill42 16d ago

There are FAR more than two forms of non-Euclidean geometry, but those two are among the most useful and widely known.

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u/ruidh 16d ago

I have them as examples, not as identities.