Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/207

Rh And, because the angle DEF is equal to the angle GEF, and the angle GEF is equal to the angle ABC, [Constr.

therefore the angle ABC is equal to the angle DEF. [Ax. 1.

For the same reason, the angle ACB is equal to the angle DFE, and the angle at A is equal to the angle at D.

Therefore the triangle ABC is equiangular to the triangle DEF.

Wherefore, if the sides &c.

PROPOSITION 6. THEOREM.

If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles another, and shall have those angles equal which are opposite to the homologous sides.

Let the triangles ABC, DEF have the angle BAC in the one, equal to the angle EDF in the other, and the sides about those angles proportionals, namely, BA to AC as ED is to DF: the triangle ABC shall be equiangular to the triangle DEF, and shall have the angle ABC equal to the angle DEF, and the angle ACB equal to the angle DFE.

At the point D, in the straight line DF, make the angle FDG equal to either of the angles BAC, EDF ; and at the point F, in the straight line DF, make the angle DFG equal to the angle ACB; [I. 23.

therefore the remaining angle at G is equal to the remain- ing angle at B.

Therefore the triangle ABC is equiangular to the triangle DGF;

therefore BA is to AC as GD is to DF. [VI. 4.

But BA is to AC as ED is to DF; [Hypothesis.

therefore ED is to DF as GD is to DF; [V. 11.

therefore ED is equal to GD. [V. 9.