Euclid's Elements Book I

1. Let AB, CD be equal and parallel, and let the straight lines AC, BD join them at the extremities in the same directions. The claim: AC, BD are also equal and parallel. (Try dragging A, B, C.)
2. Let BC be joined.

   Postulate 1

   Postulate 1: To draw a straight line from any point to any point.
3. Since AB is parallel to CD, and BC has fallen upon them, the alternate angles ABC, BCD are equal to one another.

   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. Since AB is equal to CD and BC is common, and the angle ABC is equal to the angle BCD, the base AC is equal to the base BD, the triangle ABC is equal to the triangle DCB, and the angle ACB is equal to the angle CBD.

   Proposition I.4

   Proposition I.4: 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.
5. Since the straight line BC falling on the two straight lines AC, BD makes the alternate angles ACB, CBD equal, AC is parallel to BD. And AC was also proved equal to BD.

   Proposition I.27

   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.

Being what it was required to prove.