Euclid's Elements Book I

1. With the straight line AB, and at the point B on it, let the two straight lines BC, BD, not lying on the same side, make the adjacent angles ABC, ABD equal to two right angles. The claim: BD is in a straight line with CB. (Drag the points: D always lands on CB produced.)
2. If BD is not in a straight line with CB, let BE be in a straight line with CB. (E is drawn as this assumption would place it.)
3. Since the straight line AB stands on the straight line CBE, the angles ABC, ABE are equal to two right angles.

   Proposition I.13

   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.
4. But the angles ABC, ABD are also equal to two right angles, and all right angles are equal; therefore the angles CBA, ABE are equal to the angles CBA, ABD.

   Postulate 4Common Notion 1

   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.
5. Let the angle CBA be subtracted from each; then the remaining angle ABE is equal to the remaining angle ABD, the less to the greater, which is impossible. So BE is not in a straight line with CB, and in the same way no other straight line except BD is. Therefore CB is in a straight line with BD.

   Common Notion 3Common Notion 5

   Common Notion 3: If equals be subtracted from equals, the remainders are equal.

   Common Notion 5: The whole is greater than the part.

Being what it was required to prove.