Euclid's Elements Book I

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

Proposition I.27 · Theorem

If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let the straight line EF falling on the two straight lines AB, CD make the alternate angles AEF, EFD equal to one another. The claim: AB is parallel to CD. (Try dragging E, F or B.)
  2. If not, AB, CD when produced will meet either in the direction of B, D or towards A, C. Let them meet in the direction of B, D at G. (G is drawn as this assumption would place it.)
  3. Then in the triangle GEF the exterior angle AEF is equal to the interior and opposite angle EFG, which is impossible.
    Proposition I.16
    Proposition I.16: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.
  4. Similarly they do not meet towards A, C either. Straight lines which do not meet in either direction are parallel; therefore AB is parallel to CD.
    Definition 23
    Definition 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.