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Math 409/409G History of Mathematics

Math 409/409G History of Mathematics. Book I of the Elements Part IV: Angles. In this lesson we will prove some of Euclid’s propositions about angles. But first, we must give a definition. Vertical and Adjacent Angles. Vertical angles 1 & 4 2 & 3

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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 IV: Angles

  2. In this lesson we will prove some of Euclid’s propositions about angles. But first, we must give a definition.

  3. Vertical and Adjacent Angles Vertical angles 1 & 4 2 & 3 Adjacent angles 1 & 2 1 & 3 2 & 4 3 & 4

  4. The sum of the measures of two adjacent angles is 180°. (P1.13) Given: Lines AB and CD intersect at E. Prove: AEC+CEB = 180o.

  5. Construct FE to AB at E. (P1.10) • Then FEA+ FEB = 180°. (Def ) • FEA+ FEB = FEA+ FEC+CEB = AEC+CEB(CN 2) • So AEC+ CEB = 180°. (CN 1)

  6. This proves that the sum of adjacent angles is 180o.

  7. An exterior angle of angle of a triangle is greater than either opposite interior angle. (P1.16) Prove: ACD>A

  8. Construct the bisector E of AC. (P1.10) • Then . (Def. bisector/midpoint)

  9. Construct segment BE. (Ax. 1) • On ray BE, construct point F such that . (Ax. 2, P1.3)

  10. AEB=FEC. (P1.15: vertical ’s)

  11. Construct segment FC. (Ax. 1)

  12. AEB CEF. (SAS = P1.4 = Ax. 6)

  13. So A= ECF = ACF. (Def. )

  14. ACD> ACF. (CN 5) • ACD > A. (CN 1) This proves P1.16.

  15. Comment about writing proofs When labeling congruent triangles, it is a courtesy to the reader of your proof to order the letter combination of the triangles in a way that indicates why the triangles are congruent. For example:

  16. This ends the lesson on Book I of the Elements Part IV: Angles

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