---
book: 1
number: 19
id: "I.19"
kind: "theorem"
uses: ["[[book-1/proposition-5]]", "[[book-1/proposition-18]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.19"
license: "CC-BY-SA-4.0"
---

# I.19

In any triangle the greater angle is subtended by the greater side.

![I.19](figures/I-19.svg)

## Proof

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*; [[book-1/proposition-5|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*; [[book-1/proposition-18|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.

Q. E. D.