Euclid's Elements: Book I: Proposition 18

Proposition 18: In any triangle the greater side subtends the greater angle.

For let ABC be a triangle having the side AC greater than AB;

I say that the angle ABC is also greater than the angle BCA.

For, since AC is greater than AB, let AD be made equal to AB I. 3, and let BD bejoined.

Then, since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB. I. 16

But the angle ADB is equal to the angle ABD, since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB; therefore the angle ABC is much greater than the angle ACB.

Therefore etc.


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