Transitivity of Parallel Lines

Proof

(related to "Transitivity of Parallel Lines")

Let \(AB\parallel EF\) and \(EF\parallel CB\). We want to show that \(AB\parallel CB\). Construct any secant \( {GHK} \) (see Fig.):

Since \(AB\parallel EF\), the angle \(\angle{AGH}=\angle{FHG}\), according to proposition 1.29. Similarly, the angle \(\angle{FHG}=\angle{DKH}\). Therefore \(\angle{AGK}=\angle{DKG}\), and by proposition 1.27, we have that \(AB\parallel CB\).

References

[628] Casey, John: “The First Six Books of the Elements of Euclid”, http://www.gutenberg.org/ebooks/21076, 2007
[626] Callahan, Daniel: “Euclid’s ‘Elements’ Redux”, http://starrhorse.com/euclid/, 2014

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