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.

Let *ABC* and *ADE* be equal triangles having one angle equal to one angle, namely the angle *BAC* equal to the angle *DAE.*

I say that in the triangles *ABC* and *ADE* the sides about the equal angles are reciprocally proportional, that is to say, that *CA* is to *AD* as *EA* is to *AB.*

Place them so that *CA* is in a straight line with *AD.* Therefore *EA* is also in a straight line with *AB.*

Join *BD.*

Since then the triangle *ABC* equals the triangle *ADE,* and *ABD* is another triangle, therefore the triangle *ABC* is to the triangle *ABD* as the triangle *ADE* is to the triangle *ABD.*

But *ABC* is to *ABD* as *AC* is to *AD,* and *ADE* is to *ABD* as *AE* is to *AB.*

Therefore also *AC* is to *AD* as *AE* is to *AB.*

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

Next, let the sides of the triangles *ABC* and *ADE* be reciprocally proportional, that is to say, let *AE* be to *AB* as *CA* is to *AD.*

I say that the triangle *ABC* equals the triangle *ADE.*

If *BD* is again joined, since *AC* is to *AD* as *AE* is to *AB,* while *AC* is to *AD* as the triangle *ABC* is to the triangle *ABD,* and *AE* is to *AB* as the triangle *ADE* is to the triangle *ABD,* therefore the triangle *ABC* is to the triangle *ABD* as the triangle *ADE* is to the triangle *ABD.*

Therefore each of the triangles *ABC* and *ADE* has the same ratio to *ABD.*

Therefore the triangle *ABC* equals the triangle *ADE.*

Therefore, *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.*

Q.E.D.