Proposition: 1.28: Parallel Lines II

(Proposition 28 from Book 1 of Euclid's “Elements”)

If a straight line falling across two straight lines makes the external angle equal to the internal and opposite angle on the same side, or (makes) the (sum of the) internal (angles) on the same side equal to two right angles, then the (two) straight lines will be parallel to one another.


Modern Formulation

If a straight line \(EF\) intersects two straight lines \(AB\), \(CD\) at one and only point point each such that the exterior angle \(\angle{BGE}\) equals its corresponding interior angle \(\angle{DHG}\) or if it makes two interior angles on the same side \(\angle{HGB}, \angle{DHG}\) equal to two right angles, then the two lines are parallel \((AB\parallel CD)\).

Tha angles \(\angle{BGE}\) and \(\angle{DHG}\) are called corresponding angles.

Proofs: 1

Proofs: 1 2 3 4 5 6
Propositions: 7

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"