Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/208

184 And DF is common to the two triangles EDF, GDF therefore the two sides ED,DF are equal to the two sides GD, DF each to each;

and the angle EDF is equal to the angle GDF; [Constr. therefore the base EF is equal to the base GF, and the triangle EDF to the triangle GDF, and the remaining angles to the remaining angles, each to each, to which the equal sides are opposite; [I. 4. therefore the angle DFG is equal to the angle DFE, and the angle at G is equal to the angle at E. But the angle DFG is equal to the angle ACB; [Constr. therefore the angle ACB is equal to the angle DFE. [Ax. 1. And the angle BAC is, equal to the angle EDF; [Hypothesis. therefore the remaining angle at B is equal to the remaining angle at E. Therefore the triangle ABC is equiangular to the triangle DEF.

Wherefore, if two triangles &c.

PROPOSITION 7. THEOREM.

If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then if each of the remaining angles be either less, or not less, than a right angle, or if one of them he a right angle, the triangles shall he equiangular to one another, and shall have those angles equal about touch the sides are proportionals.

Let the triangles ABC, DEF have one angle of the one equal to one angle of the other, namely, the angle BAC equal to the angle EDF, and the sides about two other angles ABC, DEF, proportionals, so that AB is to BC as DE is to EF; and, first, let each of the remaining angles at C and F be less than a right angle: the triangle ABC shall be equiangular to the triangle DEF and shall