Proposition: 1.27: Parallel Lines I

(Proposition 27 from Book 1 of Euclid's “Elements”)

If a straight line falling across two straight lines makes the alternate angles equal to one another then the (two) straight lines will be parallel to one another. * For let the straight line $EF$, falling across the two straight lines $AB$ and $CD$, make the alternate angles $AEF$ and $EFD$ equal to one another. * I say that $AB$ and $CD$ are parallel.


Modern Formulation

If a straight line \((EF)\) intersects two straight lines \((AB)\), \((CD)\) such that the alternate angles are equal \((\angle{AEF}=\angle{DFE})\), then these lines are parallel \((AB\parallel CD)\).

Proofs: 1

Proofs: 1 2 3 4 5
Propositions: 6

Thank you to the contributors under CC BY-SA 4.0!



Adapted from CC BY-SA 3.0 Sources:

  1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

  1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"