the greater angle of a triangle is subtended by the greater side
I.19
In any triangle the greater angle is subtended by the greater side.
—Euclid
Let ABC be a triangle having the angle ABC greater than the angle BCA;
I say that the side AC is also greater than the side AB.
For, if not, AC is either equal to AB or less.
Now AC is not equal to AB; for then the angle ABC would also have been equal to the angle ACB; [I.5] but it is not;
therefore AC is not equal to AB.
Neither is AC less than AB, for then the angle ABC would also have been less than the angle ACB; [I.18] but it is not;
therefore AC is not less than AB.
And it was proved that it is not equal either.
Therefore AC is greater than AB.
Therefore etc.
references
[I.5]: /elem.1.5 “Book 1 - Proposition 5” [I.18]: /elem.1.18 “Book 1 - Proposition 18”