Euclid's Elements Book I

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

Proposition I.8 · Theorem

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

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 base BC equal to the base EF. The claim: the angle BAC is equal to the angle EDF. (Drag F to turn the triangle DEF.)
  2. Let the triangle ABC be applied to the triangle DEF, with the point B placed on the point E and the straight line BC on EF. Since BC is equal to EF, the point C will coincide with F.
  3. BC coinciding with EF, BA and AC will also coincide with ED and DF. Otherwise they would fall beside them, as EG, GF; then on the same straight line EF, on the same side, there would be two other straight lines meeting at another point and equal to the two drawn from the same ends, which is impossible.
    Proposition I.7
    Proposition I.7: Given two straight lines constructed on a straight line from its extremities and meeting in a point, there cannot be constructed on the same straight line from its extremities, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively.
  4. Therefore the angle BAC coincides with the angle EDF and is equal to it.
    Common Notion 4
    Common Notion 4: Things which coincide with one another are equal to one another.