Proposition: 1.15: Opposite Angles on Intersecting Straight Lines

(Proposition 15 from Book 1 of Euclid's “Elements”)

If two straight lines cut one another then they make the vertically opposite angles equal to one another. * For let the two straight lines $AB$ and $CD$ cut one another at the point $E$. * I say that angle $AEC$ is equal to (angle) $DEB$, and (angle) $CEB$ to (angle) $AED$.


Modern Formulation

If two straight lines \(AB\) and \(CD\) intersect one another at one point \(E\), their opposite angles are equal.

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8

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"