Because the angle EFG is greater than the angle DFG, Axiom 9. and the angle DFG is greater than the angle EGF; Proved. therefore the angle EFG is greater than the angle EGF.

V.

Because the angle EFG is greater than the angle EGF; Proved. therefore EG is greater than EF.

I. 19.

Proposition XXIII. At a given point in a given straight line a rectilineal angle may be made equal to a given rectilineal angle.

Proposition III. From a given point a straight line may be drawn equal to a given straight line.

Postulate 1. A straight line may be drawn from any one point to any other point.

Proposition IV. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by these sides equal, the third sides are equal.

Proposition V. The angles at the base of an isosceles triangle are equal to one another.

Axiom 9. The whole is greater than its part.

Proposition XIX. The greater angle of every triangle has the greater side opposite to it.

Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another.

If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle contained by the sides of that which has the greater base, shall be greater than the angle contained by the sides, equal to them, of the other.

In the triangles ABC and DEF;

GIVEN that AB is equal to DE, AC equal to DF, and BC greater than EF;

IT IS REQUIRED TO PROVE that the angle BAC is greater than the angle EDF.

Because if the angle BAC were equal to the angle EDF,

BC would be equal to EF,

and BC is greater than EF;

Hyp. and I. 4.

Hyp.

G.

therefore the angle BAC is not equal to the angle EDF.

Because if the angle BAC were less than the angle EDF,

BC would be less than EF, and BC is greater than EF;

Hyp. and I. 24.

therefore the angle BAC is not less than the angle EDF.

Because the angle BAC is not equal to, nor less than,
the angle EDF;

therefore the angle BAC is greater than the angle EDF.

Hyp.

G.

Proved.

0.

Therefore, if two triangles &c.

Q. E. D.

Proposition IV. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those two sides equal, the third sides are equal.

Proposition XXIV. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides, equal to them, of the other, the base of that which has the greater angle is greater than the base of the other.

If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each, then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

Case I.

In the triangles ABC and DEF;

GIVEN that the angle ABC is equal to the angle DEF, and the angle BCA equal to the angle EFD, and that BC is equal to EF;

IT IS REQUIRED TO PROVE that the angle BAC is equal to the angle EDF, and that AB is equal to DE, and AC to DF.

Supposing AB not equal to DE, let AB be the greater.

From BA cut off BG equal to ED.

Join CG.

m.

I. 3.

Post. 1.

Because if AB is not equal to DE, then the angle ACB is equal to

the angle GCB,

and this is impossible; therefore AB is equal to DE.

Proved. Axiom 9.

G.

In the triangles ABC and DEF;

Proved.

Нур. Hyp.

}

I. 4.

because AB is equal to DE,

and BC is equal to EF,

and the angle ABC is equal to the angle DEF;

therefore AC is equal to DF,

and the angle BAC is equal to the angle EDF.

Proposition III. From the greater of two given straight lines a part may be cut off equal to the less.

Postulate 1. A straight line may be drawn from any one point to any other point.

Proposition IV. If two triangles have two sides of the one equal to two sides of the other each to each, and have also the angles contained by those sides equal to one another, their third sides are equal and their other angles are equal, namely, those to which the equal sides are opposite.

Axiom 1. Magnitudes which are equal to the same magnitude are equal to one another.

Axiom 9. The whole is greater than its part.