Euclid,

ElementsI 30©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 30: The parallels to the same straight-line are also parallel to one another.

(diagram 1) Let each of AB, GD, be parallel to EZ. I say that AB is also parallel to GD. (diagram 2 = general diagram) For let a straight-line, HK, fall into them. (diagram 3) And since a straight-line, HK, falls into parallel straight-lines, AB, EZ, therefore, the angle by AHK is equal to that by HQZ. (diagram 4) Again, since a straight-line, HK, falls into parallel straight-lines, EZ, GD, the angle by HQZ is equal to that by HKD.But the angle by AHK was also shown equal to that by HQZ. Therefore, the angle by AHK is also equal to that by HKD. And they are alternate. Therefore, AB is parallel to GD. Therefore, the parallels to the same straight-line are also parallel to one another, just what it was required to show.

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