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The Measure of Angles and Segments

    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 (Measure Of Angles) There is a unique way of assigning a degree measure to each angle such that the following properties hold:
    (i) is a real number such that
    (ii) if and only if is a right angle.
    (iii) if and only if  
    (iv) If is interior to then
    (v) For every real number between 0 and 180, there exists an angle such that
    (vi) If is supplementary to then
    (vii) if and only if

Proposition (Measure Of Segments) Given a segment called a unit segment, there is a unique way of assigning a length to each segment such that following properties hold:
    (i) is a positive real number and
    (ii) if and only if
    (iii) if and only if  
    (iv) if and only if
    (v) For every positive real number there exists a segment such that

Definition (Acute and Obtuse Angles) Using degree notation is defined as acute if and is defined as obtuse if  

Proposition (Two Angles In A Triangle) The sum of the degree measures of any two angles of a triangle is less than

    Proof. Let be given with the supplement of By the Measure Of Angles Proposition, that is, and so by the Exterior Angle Proposition Whence

Proposition (Triangle Inequality) If and are three noncolinear points, then

    Proof. By the Segment Shift Axiom there is a point such that and By the Pappus Property, By the Larger Angle Larger Side Proposition and by the Measure of Segments Proposition, it follows that by substitution. By the Interior Of An Angle Proposition is between and and thus by the Angle Relation Definition By the Ordering of Angle Proposition, and thus by the Larger Angle Larger Side Proposition Whence

Proposition (Equivalent Angle Sum) Let be the midpoint of and the unique point on such that and Then has the same angle sum of and either or is less than or equal to

    Proof. By SAS and the Special Angles Proposition, and thus by the Measure Of Angles Proposition,   By the Bisectors Proposition, either is less than the bisector of or is less than the bisector of or one of them is the bisector of In any case, either or is less than or equal to   by the Measure of Angles Proposition.

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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