## Euclid, Elements of Geometry, Book I, Proposition 18 (Edited by Dionysius Lardner, 1855)

Proposition XVIII. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]

 (96) In any triangle (B A C) if one side (A C) be greater than another (A B), the angle opposite to the greater side is greater than the angle opposite to the less.

From the greater side A C cut off the part A D equal to the less (III), and conterminous with it, and join B D.

The triangle B A D being isosceles (V), the angles A B D and A D B are equal; but A D B is greater than the internal angle A C B (XVI): therefore A B D is greater than A C B, and therefore A B C is greater than A C B: but A B C is opposite the greater side A C, and A C B is opposite the less A B.

This proposition might also be proved by producing the lesser side A B, and taking A E equal to the greater side. In this case the angle A E C is equal to A C E (V), and therefore greater than A C B. But A B C is greater than A E C (XVI), and therefore A B C is greater than A C B

Next: Proposition 19

Previous: Proposition 17

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