Euclid's Elements Book I

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

Proposition I.30 · Theorem

Straight lines parallel to the same straight line are also parallel to one another.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let each of the straight lines AB, CD be parallel to EF. The claim: AB is also parallel to CD. (Try dragging the points.)
  2. Let the straight line GK fall upon them.
  3. Since the straight line GK falls on the parallel straight lines AB, EF, the angle AGK is equal to the angle GHF.
    Proposition I.29
    Proposition I.29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
  4. Again, since the straight line GK falls on the parallel straight lines EF, CD, the angle GHF is equal to the angle GKD.
    Proposition I.29
    Proposition I.29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
  5. Therefore the angle AGK is also equal to the angle GKD; and they are alternate. Therefore AB is parallel to CD.
    Common Notion 1Proposition I.27
    Common Notion 1: Things which are equal to the same thing are also equal to one another.
    Proposition I.27: 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.