Proposition: 1.25: Angles and Sides in a Triangle IV

(Proposition 25 from Book 1 of Euclid's “Elements”)

If two triangles have two sides equal to two sides, respectively, but (one) has a base greater than the base (of the other), then (the former triangle) will also have the angle encompassed by the equal straight lines greater than the (corresponding) angle (in the latter).


Modern Formulation

If two triangles (\(\triangle{ABC}\), \(\triangle{DEF}\)) have two sides of one triangle (\(\overline{AB}\), \(\overline{AC}\)) respectively equal to two sides of the other (\(\overline{DE}\), \(\overline{DF}\)), where the base of one \((\overline{BC})\) is greater than the base of the other \((\overline{EF})\), then the angle \((\angle{BAC})\) contained by the sides of the triangle with the longer base \((\triangle{ABC})\) is greater in measure than the angle \((\angle{EDF})\) contained by the sides of the other triangle \((\triangle{DEF})\).

In shorter words, if \(\overline{AB}=\overline{DE}\), \(\overline{AC}=\overline{DF}\), and \(\overline{BC} > \overline{EF}\), then \(\angle{BAC} > \angle{EDF}\).1

Proofs: 1 Corollaries: 1

Proofs: 1 2 3

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"


  1. Note: This is a conversion of proposition 1.24