# Biangles

• George E. Martin
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

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.

## Keywords

Critical Angle Acute Angle Euclidean Geometry Hyperbolic Plane Hyperbolic Geometry
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