Her følger beviset for Euklids sætning 2
Vinkelsummen i en trekant er lig med to rette.
Bevis
Trin 1
Vi skal bevise, at vinkelsummen ( ∠ A + ∠ B + ∠ C) i en trekant er lig med to rette vinkler eller 180 grader.
Vi tegner gennem C en linje, som er parallel med AB. Derefter forlænger vi AC og BC ud over C. Nu har vi fået 3 nye vinkler x, y og z omkring C.
Trin 2
Euklids 3 almene love siger
- Det hele er lig summen af dets dele.
- Hvis man lægger det samme til lige store størrelser, fås lige store størrelser. I moderne symbolsprog: a = b medfører a + c = b + c.
- Hvis man trækker det samme fra lige store størrelser, fås lige store størrelser. I moderne symbolsprog: a = b medfører a – c = b – c.
Ifølge almene lov 1, så er det hele altså lig med summen af delene.
Og det hele (∠ x + ∠ y + ∠ z) er ifølge Euklids definition 2 lig med 180 grader, da “når en halvlinje er oprejst på en linje, så de ved siden af hinanden liggende vinkler er lige store, er enhver af de lige store vinkler ret.” Dermed ved vi, at der sammenlagt er 180 grader eller summen af 2 rette på den ene side af en linje og ∠ x + ∠ y + ∠ z = 180.
Trin 3
Euklids Aksiom 4 siger, at “Hvis to parallelle linjer skæres af en tredje, så er de ensliggende vinkler lige store.”
Dermed ved vi nu, at ∠ x =∠ B og ∠ z = ∠ A, da de er ensliggende vinkler på en tredje linje, som skærer 2 parallelle linjer (AB og den parallele linje gennem C).
Trin 4
Euklids sætning 1 siger, at “Topvinkler er lige store.“.
Og dermed er ∠ y = ∠ C, da de er topvinkler.
Geogebra
Sætning 1 – ggbm.at/PNZftMhC
Sætning 2 – ggbm.at/WACJetNh
Aksiom 4 – ggbm.at/yFJEMPRU
Litteratur
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apa
creator
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