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Math 409/409G History of Mathematics

Math 409/409G History of Mathematics. Book I of the Elements Part II.

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Math 409/409G History of Mathematics

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  1. Math 409/409GHistory of Mathematics Book I of the Elements Part II

  2. Euclid’s fifth proposition states that the base angles of an isosceles triangle are equal. This proposition was also called “the bridge of fools” because the diagram in Euclid’s proof resembled a bridge and because many weaker geometry students could not follow the logic of the proof and thus could not cross over to the rest of the Elements.

  3. But we know there is an easier proof; simply reverse the triangle and compare the two resulting triangles. StatementReason Def. isosceles  Identity Ax. 6 (SAS) Def. congruence

  4. This proves Proposition #5: The base angles of an isosceles triangle are equal.

  5. Euclid’s congruence theorems SAS (Proposition 1.4, Axiom 6): Two triangles are congruent if two sides and the included angle of one are respectively equal to two sides and the included angle of the other triangle.

  6. SSS (Proposition 1.8) Two triangles are congruent if three sides of one are respectively equal to three sides of the other.

  7. ASA (Proposition 1.26, part 1) Two triangles are congruent if two angles and the side joining these angles of one triangle are respectively equal to two angles and the side joining those angles of the other triangle.

  8. AAS (Proposition 1.26, part 2) Two triangles are congruent if two angles of one are respectively equal to two angles of the other and if a side subtending one of these angles in the first triangle is equal to the respective side subtending the same angle in the other triangle.

  9. Euclid used the technique of “proof by contradiction” to prove SSS, ASA, and AAS. Here’s an outline of how he used this technique to prove ASA.

  10. Given: Prove: By way of contradiction assume that Assume that By P1.3 there is a point G on EF such that

  11. By SAS, ABC  DEG. So by the def. of , A EDG. And by CN 5, EDG < D. Thus by CN 1, A< D.

  12. But A< D contradicts the hypothesis that A D. So the assumption that can’t be true. Thus it must be true that It now follows from SAS that the triangles are congruent.

  13. In addition to SAS, SSS, ASA, and AAS, there are two other possible configurations, namely SSA and AAA. Can either of these result in congruent triangles?

  14. Clearly, AAA does not result in congruent triangles. This configuration does result in similar triangles, a topic Euclid covered in Book VI of the Elements.

  15. Is SSA a congruence? The answer to this question is left to you as an assignment. You must either prove that it is true or produce a counterexample showing that it is not true.

  16. This ends the lesson on Book I of the Elements Part II

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