The Thirteen Books of the Elements, Vol. 2 by Euclid

Author:Euclid
Language: eng
Format: epub
Publisher: Dover Publications

For, if the angle ABC is unequal to the angle DEF, one of them is greater.

Let the angle ABC be greater ; and on the straight line AB, and at the point B on it, let the angle ABG be constructed equal to the angle DEF. [I. 23]

Then, since the angle A is equal to D, and the angle ABG to the angle DEF, therefore the remaining angle AGB is equal to the remaining angle DFE. [I. 32]

Therefore the triangle ABG is equiangular with the triangle DEF.

Therefore, as AB is to BG, so is DE to EF [VI. 4]

But, as DE is to EF, so by hypothesis is AB to BC therefore AB has the same ratio to each of the straight lines BC, BG; [V. II]

therefore BC is equal to BG, [V. 9]

so that the angle at C is also equal to the angle BGC. [I. 5]

But, by hypothesis, the angle at C is less than a right angle; therefore the angle BGC is also less than a right angle; so that the angle AGB adjacent to it is greater than a right angle. [I. 13]

And it was proved equal to the angle at F; therefore the angle at F is also greater than a right angle.

But it is by hypothesis less than a right angle: which is absurd.

Therefore the angle ABC is not unequal to the angle DEF; therefore it is equal to it.

But the angle at A is also equal to the angle at D; therefore the remaining angle at C is equal to the remaining angle at F. [I. 32]

Therefore the triangle ABC is equiangular with the triangle DEF.

But, again, let each of the angles at C, F be supposed not less than a right angle; I say again that, in this case too, the triangle ABC is equiangular with the triangle DEF.

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