Is Playfair's Axiom really a theorem? If not, why exactly?

I am trying to teach myself geometry and I need help with this question.

At first glance, it appeared to me that there is a rather obvious argument to prove the truth of Playfair's Axiom (meaning, of course, that it would be really a theorem, and not an axiom after all).

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Let AB and CD be parallel line segments. Let E be a point on AB, and F a point on CD, and EF the transversal drawn between them.

Let line GH be drawn so as to intersect AB at E.

[Assuming points A and G are on the same side as you look at the figure,] let a line be drawn connecting A with G to form the triange AEG.

Since every interior angle of a triangle has a definite magnitude, angle GEF cannot be equal to angle AEF, but must be either greater or smaller (depending on which way you have oriented line GH).

But if angle GEF is not equal to angle AEF, then the sum of angles GEF and CFE and the sum of angles AEF and CFE cannot both be equal to two right angles.

Therefore, lines AB and GH cannot both be parallel to line CD.

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Now that I have written this out, it occurs to me that insofar as it is valid, it would prove that there cannot be MORE than one parallel; arguably, it might not be enough to establish that there must be exactly one (as required by Playfair's Axiom).

By way of comment, I don't think the above argument presupposes Playfair's Axiom. Euclid makes lots of use of parallel lines, and he evidently thought this was justifiable in terms of his own Definitions and Postulates, without ever invoking a postulate that we would recognise as Playfair's Axiom (to the best of my knowledge, anyway).

I've looked through The Elements and, if Euclid anywhere deals with this specific question, I must have missed it; for which I am ready to apologise with a suitably red face... otherwise, I would appreciate any comments.