PROPOSITION 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.
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. [
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. [
V. 7]
But, as
CAB is to
BAD, so is
CA to
AD, [
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. [
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, [
VI. 1] therefore, as the triangle
ABC is to the triangle
BAD, so is the triangle
EAD to the triangle
BAD. [
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. [
V. 9]
Therefore etc. Q. E. D.