RELFE BROTHERS' EUCLID SHEETS— Props. 1-26, Book I, are now published in a similar form to this. , PROPOSITION XXVI. 164. 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, viz., either the sides adjacent to the equal angles in each, or the sides opposite to them; then shail the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other. Let be two triangles which have the angles equal to the angles each to each, namely, , and to ; also one side equal to one side. First, let those sides be equal which are adjacent to the angles that are equal in the two triangles, namely, to Then the other sides shall be equal, each to each, namely, to and to and the third angle to the third angle to For, if be greater than make equal to and join because is equal to and to the two sides, each to each; and the angle is equal to the angle ; therefore the base is equal to the base and the triangle to the triangle and the other angles to the other angles, each to each, to which the equal sides are opposite; therefore the angle is equal to the angle ; but the angle is, by the hypothesis, equal to the angle ; wherefore also the angle is equal to the angle ; the less angle equal to the greater, which is impossible; therefore is not unequal to that is, is equal to Hence, in the triangles is equal to and to and the angle is equal to the angle ; therefore the base is equal to the base and the third angle to the third angle Secondly, let the sides which are opposite to one of the equal angles in each triangle be equal to one another, namely, equal to Then in this case likewise the other sides shall be equal, to and to and also the third angle to the third angle ; because ; make For if be not equal to , one of them must be greater than the other. If possible, let be greater than equal to because is equal to and to and the angle to the angle therefore the base i is equal to the base and the triangle to the triangle and the other angles to the other angles, each to each, to which the equal sides are opposite; therefore the angle is equal to the angle ; but the angle is equal to the angle ; therefore the angle is equal to the angle that is, the exterior angle of the triangle is equal to its interior and opposite angle ; which is impossible; wherefore is not unequal to that is, is equal to is equal to Hence, in the triangles and to and the included angle is equal to the included angle ; therefore the base is equal to the base and the third angle to the third angle Wherefore, if two triangles, &c. 7831 01 ; because RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1–26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XXV. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of one greater than the base of the other ; the angle contained by the sides of the one which has the greater base, shall be greater than the angle contained by the sides, equal to them, of the other. For if the angle be not greater than the angle it must either be equal to it, or less than it. If the angle were equal to the angle then the base would be equal to the base ; ; but it is not equal, therefore the angle is not equal to the angle Again, if the angle were less than the angle then the base would be less than the base ; but it is not less, therefore the angle is not less than ; and it has been shewn, that the angle is not equal to the angle ; therefore the angle is greater than the angle Wherefore, if two triangles, &c. the angle RELFE BROTHERS' EUCLID SHEETS. 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, shall be greater than the base of the other. Let the two be two triangles, which have the two sides equal to each to each, namely, equal to and to ; but the greater than the angle Then the base shall be greater than the angle base 7 , let ; make or ; Of the two sides be not greater than ; at the point , in the line and on the same side of it as make the angle equal to the angle equal to and join Then, because is equal to and to are equal to the two therefore the base is equal to the base And because is equal to in the triangle therefore the angle is equal to the angle ; is greater than the angle ; therefore the angle is also greater than the angle ; much more therefore is greater than the angle And because in the triangle the angle is greater than the angle and that the greater angle is subtended by the greater side; therefore the side is greater than the side was proved equal to ; therefore is greater han Wherefore, if two triangles, &c. che angle ; but |