[It is easy to see, as in the case of Proposition iv., that an immediate proof of this Proposition can also be got from Proposition ii.].
Cor. 1.—If the ratio of two sides of a triangle be given, and the angle between them, the triangle is given in species.
PROP. VII.—Theorem.
If two triangles (ABC, DEF) have one angle (A) one equal to one angle (D) in the other, the sides about two other angles (B, E) proportional (AB : BC :: DE : EF), and the remaining angles (C, F) of the same species (i. e. either both acute or both not acute), the triangles are similar.
Dem.—If the angles B and E are not equal, one must be greater than the other. Suppose ABC to be the greater, and that the part ABG is equal to DEF, then the triangles ABG, DEF have two angles in one equal to two angles in the other, and are [I. xxxii.] equiangular.
| Therefore | AB : BG | :: DE : EF [iv.]; | |||||
| but | AB : BC | :: DE : EF (hyp.). |
Therefore BG is equal to BC. Hence the angles BCG, BGC must be each acute [I. xvii.]; therefore AGB must be obtuse; hence DFE, which is equal to it, is obtuse; and it has been proved that ACB is acute; therefore the angles ACB, DFE are of different species; but (hyp.) they are of the same species, which is absurd. Hence the angles B and E are not unequal, that is, they are equal. Therefore the triangles are equiangular.
Cor. 1.—If two triangles ABC, DEF have two sides in one proportional to two sides in the other, AB : BC :: DE : EF, and the angles A, D opposite one pair of homologous sides equal, the angles C, F opposite the other are either equal or supplemental. This Proposition is nearly identical with vii.
Cor. 2.—If either of the angles C, F be right, the other must be right.