The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 1. Book I, Proposition 29 is the first proposition in the Elements that requires the parallel postulate. While students taking geometry in high school likely won't see this proof, they'll be familiar with the Z, F, and C angle properties proven here.

Book I | Book II | Book IX |
---|---|---|

Definitions, Postulates, and Common Notions | Definitions | Proposition 20 |

Proposition 1, Proposition 3, | Proposition 14 | Proposition 36 |

Proposition 5, Proposition 6, | ||

Proposition 29, Proposition 47 |

*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.*

For let the straight line `EF` fall on the parallel straight lines `AB`, `CD`;

I say that it makes the alternate angles `AGH`, `GHD` equal, the exterior angle `EGB` equal to the interior and opposite angle `GHD`, and the interior angles on the same side, namely `BGH`, `GHD`, equal to two right angles.

For, if the angle `AGH` is unequal to the angle `GHD`, one of them is greater.

Let the angle `AGH` be greater.

Let the angle `BGH` be added to each;

therefore the angles `AGH`, `BGH` are greater than the angles `BGH`, `GHD`.

But the angles `AGH`, `BGH` are equal to two right angles;
[I. 13]

therefore the angles `BGH`, `GHD` are less than two right angles.

But straight lines produced indefinitely from angles less than two right angles meet; [Post. 5]

therefore `AB`, `CD`, if produced indefinitely, will meet; but they do not meet, because they are by hypothesis parallel.

Therefore the angle `AGH` is not unequal to the angle `GHD`,

and is therefore equal to it.

Again, the angle `AGH` is equal to the angle `EGB`;
[I. 13]

therefore the angle `EGB` is also equal to the angle `GHD`.
[C.N. 1]

Let the angle `BGH` be added to each;

therefore the angles `EGB`, `BGH` are equal to the angles `BGH`, `GHD`.
[C.N. 2]

But the angles `EGB`, `BGH` are equal to two right angles;
[I. 13]

therefore the angles `BGH`, `GHD` are also equal to two right angles.

Therefore etc.Q.E.D.