Proposition XXVII. Theorem.
|(114)||If a line (E F) intersect two right lines (A B and C D), and make the alternate angles equal to each other (A E F to E F D), these right lines are parallel.|
For, if it be possible, let those lines not be parallel but meet in G; the external angle A E F of the triangle E G F is greater than the internal E F G (XVI); but it is also equal to it (by hyp.), which is absurd; therefore A B and C D do not meet at the side B D; and in the same manner it can be demonstrated, that they do not meet at the side A C; since, then, the right lines do not meet on either side they are parallel.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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