Euclid to playfair with Recording.pptx - Playfairu2019s...

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Playfair’s Axiom equivalent to Euclid’s Fifth Postulate Proofs
A Closer Look At Postulate 5 If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. These lines will meet
Proving a Statement Equivalent to the Fifth Postulate Euclid’s Elements: Book 1 Definitions, postulates, common notions, and propositions 1 -28.(Basis E). 2 Lemmas and a Theorem: Lemma 1 : Lines intersect at a unique point Lemma 2 : If l || m , then all points on l are on the same side of m . Theorem 1 : Two lines perpendicular to the same line are parallel
EV Playfair’s Axiom If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Through a point not on a given line there passes not more than one parallel to the line.

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