---
book: 6
number: 15
id: "VI.15"
kind: "theorem"
uses: ["[[book-1/proposition-14]]", "[[book-5/proposition-7]]", "[[book-6/proposition-1]]", "[[book-5/proposition-11]]", "[[book-5/proposition-9]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.15"
license: "CC-BY-SA-4.0"
---

# VI.15

*In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle*, *and in which the sides about the equal angles are reciprocally proportional*, *are equal*.

## Proof

Let *ABC*, *ADE* be equal triangles having one angle equal to one angle, namely the angle *BAC* to the angle *DAE*; I say that in the triangles *ABC*, *ADE* the sides about the equal angles are reciprocally proportional, that is to say, that, as *CA* is to *AD*, so is *EA* to *AB*.

For let them be placed so that *CA* is in a straight line with *AD*; therefore *EA* is also in a straight line with *AB*. [[book-1/proposition-14|I. 14]]

Let *BD* be joined.

Since then the triangle *ABC* is equal to the triangle *ADE*, and *BAD* is another area, therefore, as the triangle *CAB* is to the triangle *BAD*, so is the triangle *EAD* to the triangle *BAD*. [[book-5/proposition-7|V. 7]]

But, as *CAB* is to *BAD*, so is *CA* to *AD*, [[book-6/proposition-1|VI. 1]] and, as *EAD* is to *BAD*, so is *EA* to *AB*. [*id*.]

Therefore also, as *CA* is to *AD*, so is *EA* to *AB*. [[book-5/proposition-11|V. 11]]

Therefore in the triangles *ABC*, *ADE* the sides about the equal angles are reciprocally proportional.

Next, let the sides of the triangles *ABC*, *ADE* be reciprocally proportional, that is to say, let *EA* be to *AB* as *CA* to *AD*; I say that the triangle *ABC* is equal to the triangle *ADE*.

For, if *BD* be again joined, since, as *CA* is to *AD*, so is *EA* to *AB*, while, as *CA* is to *AD*, so is the triangle *ABC* to the triangle *BAD*, and, as *EA* is to *AB*, so is the triangle *EAD* to the triangle *BAD*, [[book-6/proposition-1|VI. 1]] therefore, as the triangle *ABC* is to the triangle *BAD*, so is the triangle *EAD* to the triangle *BAD*. [[book-5/proposition-11|V. 11]]

Therefore each of the triangles *ABC*, *EAD* has the same ratio to *BAD*.

Therefore the triangle *ABC* is equal to the triangle *EAD*. [[book-5/proposition-9|V. 9]]

Therefore etc. Q. E. D.