the greater side of a triangle subtends the greater angle
I.18
In any triangle the greater side subtends the greater angle.
—Euclid
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. [1]
Therefore etc.
## References
[I.3]: /elem.1.3 “Book 1 - Proposition 3” [I.16]: /elem.1.16 “Book 1 - Proposition 16”
## Footnotes