A biangle is defined in Definition 23.8. For Euclidean geometry every interior ray of a biangle intersects both sides of the biangle. However, this is not the case under the Hypothesis of the Acute Angle. For example, if ⨆PQRS has right angles at Q and R, the negotion of Play-fair’s Parallel Postulate implies that some interior ray \[\overrightarrow {QT} \] is parallel to \[\overleftrightarrow {RS}\] and so does not intersect \[\overleftrightarrow {RS}\]. If every interior ray \[\overrightarrow {BE} \] of ⨆ABCD does intersect \[\overrightarrow {CD} \], we shall say the biangle is closed from B. We shall prove that a biangle closed from one vertex is necessarily closed from the other vertex. Although trivially true for Euclidean geometry, this result is not trivial for absolute geometry.


Critical Angle Acute Angle Euclidean Geometry Hyperbolic Plane Hyperbolic Geometry 
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Copyright information

© Springer-Verlag New York, Inc. 1975

Authors and Affiliations

  • George E. Martin
    • 1
  1. 1.Department of Mathematics and StatisticsState University of New York at AlbanyAlbanyUSA

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