Book 1 - Proposition 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 sum of the interior angles on the same side equal to two right angles.

Let the straight line EF fall on the parallel straight lines AB and CD.
I say that it makes the alternate angles AGH and GHD equal, the exterior angle EGB equal to the interior and opposite angle GHD, and the sum of the interior angles on the same side, namely angle BGH and angle GHD, equal to two right angles.

If the angle AGH does not equal the angle GHD, then one of them is greater. Let the angle AGH be greater.

Add the angle BGH to each. Therefore the sum of the angles angle AGH and angle BGH is greater than the sum of the angles angle BGH and angle GHD.

But sum of the angles AGH and BGH equals two right angles. Therefore the sum of the angles BGH and GHD is less than two right angles.

But straight lines produced indefinitely from angles less than two right angles meet. Therefore line AB and line 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 therefore equals it.

Again, the angle AGH equals the angle EGB. Therefore the angle EGB also equals the angle GHD.

Add the angle BGH to each. Therefore the sum of the angles EGB and BGH equals the sum of the angles BGH and GHD.

But the sum of the angles EGB and BGH equals two right angles. Therefore the sum of the angles BGH and GHD also equals two right angles.
Therefore 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 sum of the interior angles on the same side equal to two right angles.

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