Euclid’s Fifth Postulate

May 8th, 2015
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If a straight line falling on two straight lines makes the sum of the interior angles on the same side less than the sum of two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.An equivalent axiom, known as Playfair’s Axiom, expresses Euclid’s Fifth Postulate this way: Through a point not on a given straight line, there is one and only one straight line that does not intersect the

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EsfdEuclids Fifth PostulateBy Ma. Louise De Las PeasLet us take a closer look at Euclids Fifth Postulate which asserts that:If a straight line falling on two straight lines makes the sum of the interior angles on the same side less than the sum of two right angles, then the two straight lines, if produced indenitely, meet on that side on which the angles are less than two right angles.An equivalent axiom, known as Playfairs Axiom, expresses Euclids Fifth Postulate this way: Through a point not on a given straight line, there is one and only one straight line that does not intersect the given line. Playfairs Axiom and Euclids Fifth Postulate are equivalent, because from Euclids Fifth Postulate, we can derive Playfairs Axiom. From Euclids postulates with the Playfair Axiom replacing Euclids Fifth Postulate, we can derive Euclids Fifth Postulate. The Fifth Postulate is so important, since it leads to some of the most basic, useful, and important results in Euclidean geometry. It leads to the following results: 1. The conclusion that the sum of the measure of the angles of a triangle is 1800 2. The existence of rectangles and squares 3. The existence of similar gures 4. The Pythagorean relation The Fifth Postulate aroused so much controversy among mathematicians. For 2 000 years, mathematicians tried to prove that the Fifth Postulate is a necessary consequence of Euclids other nine axioms and postulates.(The axioms and p

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