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Measure of Angles and Segments

To avoid some of the difficulties that we faced in the previous two proofs, and to facilitate matters at a later time, we will introduce a measure for angles and for segments.

The proof of the Theorem requires the axioms of continuity for the first time. The axioms of continuity are not needed if one merely wants to define the addition for congruence classes of segments and then prove the triangle inequality for these congruence classes. It is in order to prove several of our theorems that we need the measurement of angles and segments by real numbers, and for such measurement Archimedes's axiom is required. However, the fourth and eleventh parts of Theorem 11.2, the proofs for which require Dedekind's axiom, are never used in proofs in the text. It is possible to introduce coordinates without the continuity axioms, as in discussed in Appendix B of the text.

The notation tex2html_wrap_inline14900 will be used for the number of degrees in tex2html_wrap_inline11392, and the length of segment AB will be denoted by tex2html_wrap_inline14906

Theorem 10.2:  

  1. There is a unique way of assigning a degree measure to each angle such that the following properties hold:
    1. tex2html_wrap_inline14900 is a real number such that tex2html_wrap_inline14910.
    2. tex2html_wrap_inline14912 if and only if tex2html_wrap_inline11392 is a right angle.
    3. tex2html_wrap_inline14916 if and only if tex2html_wrap_inline12624.
    4. If tex2html_wrap_inline12758 is interior ot tex2html_wrap_inline14922, then tex2html_wrap_inline14924.
    5. For every real number x between 0 and 180, there exists an angle tex2html_wrap_inline11392 such that tex2html_wrap_inline14930
    6. If tex2html_wrap_inline11394 is supplementary to tex2html_wrap_inline11392, then tex2html_wrap_inline14936.
    7. tex2html_wrap_inline14938 if and only if tex2html_wrap_inline14940.
  2. Given a segment OI, called the unit segment. Then there is a unique way of assigning a length tex2html_wrap_inline14906 to each segment AB such that the following properties hold:
    1. tex2html_wrap_inline14906 is a positive real number and tex2html_wrap_inline14950.
    2. tex2html_wrap_inline14952 if and only if tex2html_wrap_inline12592.
    3. tex2html_wrap_inline12498 if and only if tex2html_wrap_inline14958.
    4. tex2html_wrap_inline14960 if and only if AB<CD.
    5. For every positive real number x, there exists a segment AB such that tex2html_wrap_inline14968.

Definition: An angle tex2html_wrap_inline11392 is acute   if tex2html_wrap_inline14972, and is obtuse   if tex2html_wrap_inline14974.

Corollary 1: The sum of the degree measures of any two angles of a triangle is less than tex2html_wrap_inline11150.

This follows from the Exterior Angle Theorem and Theorem 11.2.

Proof: We want to show that tex2html_wrap_inline14978. From the Exterior Angle Theorem and Theorem 11.2,
eqnarray2421
since they are supplementary angles.

figure2425

Corollary 2:[Triangle Inequality]  If A, B, and C are three noncollinear points, then tex2html_wrap_inline14992.

Theorem 11.2 offers an easier proof of this than the one that we gave.

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Next: Saccheri-Legendre Theorem Up: Theorems of Continuity Previous: Elementary Continuity Principle

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