Euclid's Elements Book I

1. Let the straight line EF falling on the two straight lines AB, CD make the exterior angle EGB equal to the interior and opposite angle GHD on the same side. The claim: AB is parallel to CD. (Try dragging G, H or B.)
2. Since the angle EGB is equal to its vertical angle AGH, the angle AGH is also equal to the angle GHD; and they are alternate;

   Proposition I.15Common Notion 1

   Proposition I.15: If two straight lines cut one another, they make the vertical angles equal to one another.

   Common Notion 1: Things which are equal to the same thing are also equal to one another.
3. therefore AB is parallel to CD.

   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.
4. Next, let the interior angles on the same side, BGH, GHD, be equal to two right angles.
5. The angles AGH, BGH are also equal to two right angles; therefore the angles AGH, BGH are equal to the angles BGH, GHD. Let the angle BGH be subtracted from each; then the remaining angle AGH is equal to the remaining angle GHD.

   Proposition I.13Postulate 4Common Notion 1Common Notion 3

   Proposition I.13: If a straight line set up on a straight line makes angles, it will make either two right angles or angles equal to two right angles.

   Postulate 4: That all right angles are equal to one another.

   Common Notion 1: Things which are equal to the same thing are also equal to one another.

   Common Notion 3: If equals be subtracted from equals, the remainders are equal.
6. And they are alternate; therefore AB is parallel to CD.

   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.