Euclid to playfair with Recording.pptx - Playfair’s Axiom...

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Playfair’s Axiom equivalent toEuclid’s Fifth PostulateProofs
A Closer Look At Postulate 5If a straight line falling on two straight lines makesthe interior angles on the same side less than tworight angles, the two straight lines, if producedindefinitely, meet on that side on which are theangles less than the two right angles.These lineswill meet
Proving a Statement Equivalent to the FifthPostulateEuclid’s Elements: Book 1Definitions, postulates, common notions, andpropositions 1 -28.(Basis E).2 Lemmas and a Theorem:Lemma 1: Lines intersect at a unique pointLemma 2: Ifl||m, then all points onlare on thesame side ofm.Theorem 1: Two lines perpendicular to the sameline are parallel
EVPlayfair’s AxiomIf a straight line falling on two straight linesmakes the interior angles on the same side lessthan two right angles, the two straight lines, ifproduced indefinitely, meet on that side onwhich are the angles less than the two rightangles.Through a point not on a given linethere passes not more than one parallel to theline.

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