Euclid, Elements I 31©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 31: To draw a straight line through the given point parallel to the given line.

(general diagram)

Diagrams for book 1 proposition 31, following the display, construction, and demonstration(diagram 1) Let the given point be A and the given straight-line BG. It is, in fact, required to draw a straight line through point A parallel to straight-line BG. (diagram 2) For let there be taken on BG a point as happens, D, and let AD be joined. (diagram 3) And at straight-line DA and the point on it, A, let an equal to the angle by ADG be constructed, that by DAE. (diagram 4) And let straight line AZ be extended on a straight-line with EA. And since a straight-line, AZ, falling into two straight lines, BG, EZ, has made alternate angles, those by EAD, ADG, equal to one another, therefore, EAZ is parallel to BG. Therefore, through the given point, A, parallel to the given line, BG, the straight-line EAZ has been drawn, just what it was required to make.