Proposition: 1.18: Angles and Sides in a Triangle I

(Proposition 18 from Book 1 of Euclid's “Elements”)

In any triangle, the greater side subtends the greater angle.


Modern Formulation

In a given triangle \(\triangle{ABC}\) with the side \(\overline{AC}\) longer than \(\overline{AB}\), the angle \(\angle{ABC}\) opposite to the longer side is greater than the angle \(\angle{BCA}\) opposite to the shorter side.

Proofs: 1

Proofs: 1
Propositions: 2

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"