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. * For let some straight line $AB$ stood on the straight line $CD$ make the angles $CBA$ and $ABD$. * I say that the angles $CBA$ and $ABD$ are certainly either two right angles, or (have a sum) 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"