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I am trying to show that the 30th Euclid's proposition,

"Straight lines parallel to the same straight line are also parallel to one another."

is equivalent to the 5th Postulate:

"If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles."

What happens is that if I want to prove that they are equivalent, I need to show that one implies each other.

When I started assuming the 5th postulate, I first tried to follow step by step of how this proof was build to see if I could see any logic ways to get to where I wanted do. I am just having a hard time even to get started. Any hints would be highly appreciated.

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  • $\begingroup$ This is a long proof. Did you check Greenberg's book? You can start here. $\endgroup$
    – Moishe Kohan
    May 22 at 2:53
  • $\begingroup$ The proof is not short; see original text: you need I.29, and then I.13 and I.15, and so on. And I'm sure that propositional logic will ot be enough; Common Notions are used, and this means equality, i.e. predicate logic. $\endgroup$
    – Mauro ALLEGRANZA
    May 23 at 6:43

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