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The equivalence of Side-Angle-Side and Side-Angle-Angle in the absolute plane

Abstract

Following the approach of G.D. Birkhoff, we develop a new axiom system for a continuous absolute plane. In this new axiom system we remove Side-Angle-Side as an axiom and replace it with Side-Angle-Angle as a new axiom. We prove that the new axiom system is also a continuous absolute plane, and in particular, that Side-Angle-Side holds in the new axiom system. In addition, we give new proofs of well known results. These new proofs do not depend on Side-Angle-Side, but instead use Side-Angle-Angle.

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Correspondence to John Donnelly.

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Donnelly, J. The equivalence of Side-Angle-Side and Side-Angle-Angle in the absolute plane. J. Geom. 97, 69–82 (2010). https://doi.org/10.1007/s00022-010-0038-y

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  • DOI: https://doi.org/10.1007/s00022-010-0038-y

Mathematics Subject Classification (2010)

  • 51F05

Keywords

  • Continuous absolute plane
  • foundations of geometry
  • congruence
  • triangles