1 / 16

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

Download Presentation

Math 409/409G History of Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related