---
book: 1
number: 17
id: "I.17"
kind: "theorem"
uses: ["[[book-1/proposition-16]]", "[[book-1/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.17"
license: "CC-BY-SA-4.0"
---

# I.17

In any triangle two angles taken together in any manner are less than two right angles.

![I.17](figures/I-17.svg)

## Proof

Let *ABC* be a triangle; I say that two angles of the triangle *ABC* taken together in any manner are less than two right angles.

For let *BC* be produced to *D*. [[book-1/postulates#Postulate 2|Post. 2]]

Then, since the angle *ACD* is an exterior angle of the triangle *ABC*,

it is greater than the interior and opposite angle *ABC*. [[book-1/proposition-16|I. 16]] Let the angle *ACB* be added to each; therefore the angles *ACD*, *ACB* are greater than the angles *ABC*, *BCA*. But the angles *ACD*, *ACB* are equal to two right angles. [[book-1/proposition-13|I. 13]]

Therefore the angles *ABC*, *BCA* are less than two right angles.

Similarly we can prove that the angles *BAC*, *ACB* are also less than two right angles, and so are the angles *CAB*, *ABC* as well.

Therefore etc.

Q. E. D.