What is wrong with the following proof of the SAA congruence criterion?
Consider $\triangle$ABC and $\triangle$DEF. Given $AC\cong DF$, $\angle A \cong \angle D$, and $\angle B \cong \angle E$ it follows that $\angle C \cong \angle F$ since \begin{equation} \label*{} \begin{split} (\angle C)° & = 180°-(\angle A)°-(\angle B)° \\ & = 180°-(\angle D)°-(\angle E)° \\ & = (\angle F)° \end{split} \end{equation}
In Neutral geometry, do we not know that the interior angles of a triangle add to 180°? I don't see where this proof uses Euclid's parallel postulate or its results.
The fact that the sum of the interior angles of a triangle add to 180° does not hold in Neutral geometry. It is equivalent to Euclid's parallel postulate if you assume the Archimedean axiom.