Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.19 · Theorem

In any triangle the greater angle is subtended by the greater side.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABC be a triangle having the angle ABC greater than the angle BCA. The claim: the side AC is also greater than the side AB. (Drag the vertices, keeping the angle ABC the greater.)
  2. For if not, AC is either equal to AB or less.
  3. Now AC is not equal to AB, for then the angle ABC would be equal to the angle ACB; but it is not.
    Proposition I.5
    Proposition I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.
  4. Neither is AC less than AB, for then the angle ABC would be less than the angle ACB; but it is not.
    Proposition I.18
    Proposition I.18: In any triangle the greater side subtends the greater angle.
  5. Therefore AC is greater than AB.