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I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs of an angle and make some construction based on that, but I don't really understand what that means. Any suggestions?


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What can you use for that? How is that proved in euclidean geometry, exactly? –  Berci Nov 1 '12 at 21:23
What exactly is Aristotle's Angle Unboundedness Axiom? –  Will Jagy Nov 1 '12 at 21:45
I have found it by google: For any length $AB$ and any given angle $UOV\angle$, there exists an $Y$ on one side of the angle that orthogonally projected to the other angle gives $X$ and that $XY>AB$. –  Berci Nov 1 '12 at 21:50
@Berci Yes, that is the Angle Unboundedness Axiom in Euclidean/Neutral geometry, but I need to figure out a way to prove that statement holds in Hyperbolic geometry. –  roboguy12 Nov 1 '12 at 23:29
What elementary operations are you allowed to use? I don't yet see how a general parallel line will be of any use. On the other hand, the problem would be fairly easy to solve if you were allowed curves of equal distance. In hyperbolic geometry, these are not parallel lines, not even geodesics. –  MvG Nov 6 '12 at 15:17

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