Euclid's Elements Book I

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

Proposition I.4 · Theorem

If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABC, DEF be two triangles with AB equal to DE, AC equal to DF, and the angle BAC equal to the angle EDF. The claim: the base BC is equal to the base EF, the triangle ABC is equal to the triangle DEF, and the remaining angles are equal, namely ABC to DEF and ACB to DFE. (Drag E to turn the triangle DEF.)
  2. Let the triangle ABC be applied to the triangle DEF, with the point A placed on the point D and the straight line AB on DE. Since AB is equal to DE, the point B will coincide with E.
  3. With AB on DE, the straight line AC will fall on DF, because the angle BAC is equal to the angle EDF; and since AC is equal to DF, the point C will coincide with F.
  4. B coincides with E and C with F, so the base BC coincides with the base EF (otherwise two straight lines would enclose a space, which is impossible). Things which coincide with one another are equal, so BC is equal to EF.
    Common Notion 4
    Common Notion 4: Things which coincide with one another are equal to one another.
  5. So the whole triangle ABC coincides with the whole triangle DEF and is equal to it, and the remaining angles coincide with the remaining angles and are equal to them: the angle ABC to DEF, and ACB to DFE.
    Common Notion 4
    Common Notion 4: Things which coincide with one another are equal to one another.