---
book: 6
number: 14
id: "VI.14"
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.14"
license: "CC-BY-SA-4.0"
---

# VI.14

*In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal*.

## Proof

Let *AB*, *BC* be equal and equiangular parallelograms having the angles at *B* equal, and let *DB*, *BE* be placed in a straight line; therefore *FB*, *BG* are also in a straight line. [[book-1/proposition-14|I. 14]]

I say that, in *AB*, *BC*, the sides about the equal angles are reciprocally proportional, that is to say, that, as *DB* is to *BE*, so is *GB* to *BF*.

For let the parallelogram *FE* be completed.

Since, then, the parallelogram *AB* is equal to the parallelogram *BC*, and *FE* is another area, therefore, as *AB* is to *FE*, so is *BC* to *FE*. [[book-5/proposition-7|V. 7]]

But, as *AB* is to *FE*, so is *DB* to *BE*, [[book-6/proposition-1|VI. 1]] and, as *BC* is to *FE*, so is *GB* to *BF*. [*id*.] therefore also, as *DB* is to *BE*, so is *GB* to *BF*. [[book-5/proposition-11|V. 11]]

Therefore in the parallelograms *AB*, *BC* the sides about the equal angles are reciprocally proportional.

Next, let *GB* be to *BF* as *DB* to *BE*; I say that the parallelogram *AB* is equal to the parallelogram *BC*.

For since, as *DB* is to *BE*, so is *GB* to *BF*, while, as *DB* is to *BE*, so is the parallelogram *AB* to the parallelogram *FE*, [[book-6/proposition-1|VI. 1]] and, as *GB* is to *BF*, so is the parallelogram *BC* to the parallelogram *FE*, [[book-6/proposition-1|VI. 1]] therefore also, as *AB* is to *FE*, so is *BC* to *FE*; [[book-5/proposition-11|V. 11]] therefore the parallelogram *AB* is equal to the parallelogram *BC*. [[book-5/proposition-9|V. 9]]

Therefore etc. Q. E. D.