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

../../../../_images/elem.1.prop.19.png

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”