Proposition: 1.13: Angles at Intersections of Straight Lines

(Proposition 13 from Book 1 of Euclid's “Elements”)

If a straight line stood on a(nother) straight line makes angles, it will certainly either make two right angles, or angles whose sum is) equal to two right angles.


Modern Formulation

If the straight line \(AB\) intersects the straight line \(CD\) at one and only one point (\(B\)), then either \(\angle{ABC}\) and \(\angle{DBA}\) are right angles or the sum \(\angle{ABC}+\angle{DBA}\) equals the sum of two right angles.

Proofs: 1 Corollaries: 1 2

Proofs: 1 2 3 4 5 6 7 8 9 10 11

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"