Euclid, Elements I 27©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 27: If a straight-line falling into two straight-lines makes alternate angles equal to one another, the straight-lines will be parallel to one another.

(general diagram)

Diagrams for book 1 proposition 27, following the display, construction, and demonstration(diagram 1) For let a straight line, EZ, falling into two straight-lines, AB, GD, make alternate angles, those by AEZ, EZD, equal to one another. (diagram 2) I say that AB is parallel to GD. For if not, AB, GD, on being extended will fall-together, whether on the B, D sides or on the A, G. Let them be extended and fall-together on the B, D sides at H. (diagram 3) In fact, the outside angle of triangle HEZ, that by AEZ, is equal to the interior and opposite angle, that by EZH, which is impossible. (diagram 4) Therefore, on being extended AB, GD will not fall-together on the B, D sides. (diagram 5) Similarly, in fact, it will be shown that they do not fall-together on the A, G, either. (diagram 6) But parallels are those that tall-together on neither sides. Therefore, AB is parallel to GD. therefore, if a straight-line falling into two straight-lines makes alternate angles equal to one another, the straight-lines will be parallel to one another, just what it was required to show.

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