Editing Hyperbolic geometry (section)

===Relation to Euclidean geometry===
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Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only [[axiom]]atic difference is the [[parallel postulate]].
When the parallel postulate is removed from Euclidean geometry the resulting geometry is [[absolute geometry]].
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of [[Euclid's Elements|Euclid's ''Elements'']], are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's ''Elements'' prove the existence of parallel/non-intersecting lines.

This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced.
Further, because of the [[angle of parallelism]], hyperbolic geometry has an [[absolute scale]], a relation between distance and angle measurements.