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The following statements are all equivalent in plane geometry:

(I).- Euclid's fifth postulate:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to 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

(II).-Playfair's axiom of parallel lines:

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. Which can be rephrased with: In a plane, given a line and a point not on it, there's a unique parallel line that can be drawn through the point, once it's been proved the existence of parallel lines.

(III).- Existence of similar triangles:

For any triangle, there exists a similar non-congruent triangle

I'm interested in the equivalence of the third statement with any of the first two (since there's plenty of proofs for the equivalence of those). I noticed there are two results quite useful that do not require the fifth postulate in order to be true:

  • Any exterior angle of a triangle is greater in measure than any of the interior non-adjacent angles.
  • If the alternate interior angles formed by two lines cut by a transversal, then the lines are parallel.

    Thanks in advance!

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