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Saccheri-Legendre Theorem

By David A. Smith

This topic defines a Dedekind cut and proves the Dedekind Axiom implies the Archimedian Axiom. After introducig the measure of a segment and an angle, the triangular inequality and the Saccheri-Legendre Theorem are proven.

Proposition (Saccheri-Legendre) The sum of the degree measures of the three angles of a triangle is less than or equal to

    Proof. Assume, on the contrary, that the angle sum of is greater than say where is a positive number. By the Equilvalent Angle Sum Proposition, replace with another traingle that has the same angle sum as but in which one of the angles has at most half the number of degrees as Repeat the procedure to get another triangle that has the same angle sum as   and has an angle that is one-quarter the degree measure of   The Archimedian Principle for real numbers guarantees that if this process is repeated enough times, eventually a triangle that has angle sum and with one angle with degree measure at most Thus, the sum of the degrees measures of the other two angles will be greater than or equal to contradicting the Two Angles In A Triangle Proposition.

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