# On the equivalence of Playfair's axiom to the parallel postulate

## TL;DR

This paper demonstrates that the classical equivalence between Euclid's parallel postulate and Playfair's axiom does not hold in non-SAS geometries, providing a counterexample where one holds but not the other.

## Contribution

It constructs a non-SAS geometry that models Playfair's axiom but does not satisfy the parallel postulate, challenging their equivalence.

## Key findings

- Classical equivalence fails without triangle congruence.
- A non-SAS geometry models Playfair's axiom but not the parallel postulate.
- Highlights the importance of triangle congruence in geometric axioms.

## Abstract

We show that the classical equivalence of Euclid's parallel postulate and Playfair's axiom collapses in the absence of triangle congruence. In particular, we construct a non-SAS geometry that models the Playfair axiom but not the parallel postulate.