Proposition XIV. Theorem.
|(86)||If two right lines (C B and B D) meeting another right line (A B) at the same point (B), and at opposite sides, make angles with it which are together equal to two right angles, those right angles (C B and B D) form one continued right line.|
For if possible, let B E and not B D be the continuation of the right line C B, then the angles C B A and A B E are are equal to two right angles (XIII), but C B A and A B D are also equal to two right angles, by hypothesis, therefore C B A and A B D taken together are equal to C B A and A B E; take away from these equal quantities C B A which is common to both, and A B E shall be equal to A B D, a part to the whole, which is absurd; therefore B E is not the continuation of C B, and in the same manner it can be proved, that no other line except B D is the continuation of it, therefore B D forms with B C one continued right line.
In the enunciation of this proposition, the student should be cautious not to overlook the condition that the two right lines C B and B E forming angles, which are together equal to two right angles, with B A lie at opposite sides of B A. They might form angles together equal to two right angles with B A, yet not lie in the same continued line, if as in this figure they lay at the same side of it. It is assumed in this proposition that the line C B has a production. This is however granted by Postulate 2.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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