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Attempts to Prove the Fifth Postulate
Parallels Without a Common Perpendicular
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acute angle angle of parallelism angle-sum antipodal points approaches assumption Axiom boundary parallels boundary rays called chords circle common perpendicular congruent corresponding arcs cosh defect definition direction of parallelism double elliptic geometry endpoint equation equidistant equivalent Euclid's Euclidean formula Euclidean geometry exceeds EXERCISES fact g and h geodesic arc given hemispherical triangle Hence horocycle hyperbolic geometry hypotenuse infinite interior point Lambert quadrilateral Law of Cosines line AC line g meet g metric space midpoint modified curve non-Euclidean geometry noncollinear nonopposite obtain obtuse opposite points pair parallel lines parallel postulate parallel to g perpendicular bisectors proof properties Proposition Prove Theorem radius right angles right triangle Saccheri quadrilateral Section segment joining simple arc sinh sphere spherical geometry spherical triangle standard unit subdivides summit angles tanh Theo Theorem transversal triangle ABC trilateral unique unit of length unit segment vertex vertices