Euclid Book 6 Proposition 7a
Statement
Computational Explanation
Explanations
Let , be two triangles having one angle equal to one angle, the angle to the angle , the sides about other angles , proportional, so that, as is to , so is to , and, first, each of the remaining angles at , less than a right angle; I say that the triangle is equiangular with the triangle , the angle will be equal to the angle , and the remaining angle, namely the angle at , equal to the remaining angle, the angle at .or, if the angle is unequal to the angle , one of them is greater.
Let the angle be greater; and on the straight line , and at the point on it, let the angle be constructed equal to the angle .
Then, since the angle is equal to , and the angle to the angle , therefore the remaining angle is equal to the remaining angle .
Therefore the triangle is equiangular with the triangle .
Therefore, as is to , so is to
But, as is to , so by hypothesis is to ; therefore has the same ratio to each of the straight lines , ;therefore is equal to ,so that the angle at is also equal to the angle .
But, by hypothesis, the angle at is less than a right angle; therefore the angle is also less than a right angle; so that the angle adjacent to it is greater than a right angle.
And it was proved equal to the angle at; therefore the angle at is also greater than a right angle.
But it is by hypothesis less than a right angle: which is absurd.
Therefore the angle is not unequal to the angle ; therefore it is equal to it.
But the angle at is also equal to the angle at ; therefore the remaining angle at is equal to the remaining angle at .
Therefore the triangle is equiangular with the triangle .
ABC
DEF
BAC
EDF
ABC
DEF
AB
BC
DE
EF
C
F
ABC
DEF
ABC
DEF
C
F
F
ABC
DEF
Let the angle
ABC
AB
B
ABG
DEF
Then, since the angle
A
D
ABG
DEF
AGB
DFE
Therefore the triangle
ABG
DEF
Therefore, as
AB
BG
DE
EF
But, as
DE
EF
AB
BC
AB
BC
BG
BC
BG
C
BGC
But, by hypothesis, the angle at
C
BGC
AGB
And it was proved equal to the angle at
F
F
But it is by hypothesis less than a right angle: which is absurd.
Therefore the angle
ABC
DEF
But the angle at
A
D
C
F
Therefore the triangle
ABC
DEF