How to prove Playfair's axiom or Euclid's parallel postulate without using angle measure

Question

I have only seen this proof done using angle measure.

How do you prove this without using angle measure or the fact that right angles are congruent?

Answer 1

There is no need to introduce angle measure. We use only the undefined intermediacy relation of point triplets. Also, we need the axioms describing the undefined concept of congruence making it possible to copy triangles (angles). Finally, we need the concept of perpendicularity.

See the following figure.

The definition of the relation $\beta>\alpha$ is based on that one can say that $B$ is between $A$ and $C$. $\alpha=\beta$ if $C=B$. On the right hand side figure we define perpendicularity by $\alpha=\alpha'$.

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Note that the Playfair axiom does not use angle comparisons. It is Euclid's Fifth postulate that uses the concept of angle comparison.