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 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. be two triangles which have the angles Let to , and to equal to the angles each to each, namely, 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, Then the other sides shall be equal, each to each, namely, and and the other angles to the other angles, each to each, to which the equal sides are is, by the hypothesis, equal the less angle equal to the greater, be not equal to and to the two sides, are equal to the two one of them must be greater than the other. If possible, let be greater , because each to each; and therefore the base ; is equal to the base ; but the angle is equal to the angle Secondly, let the sides which are opposite to one of the equal angles in each triangle be equal to one another, Then in this case likewise the other sides shall be equal, one of them must be greater than the other. If possible, let be greater and join and to and the angle Then in the two triangles because is ; therefore the base is equal to to which the equal sides are opposite; is equal to the angle is equal to its interior and opposite angle that is, is equal to ; because is equal to therefore the base Wherefore, if two triangles, &c. and the other angles to the other angles, each to each, ; but the angle that is, the ex ; which is impossible; and to , is equal to the base and the included and the third 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. to it, or less than it. If the angle were equal to the angle it must either be equal is would be equal to the base not equal to the angle Again, if the angle ; but it is not equal, therefore 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 the angle ; therefore the angle ; and it has been shewn, that the angle Wherefore, if two triangles, &c. is greater than the angle is not equal to the angle PROPOSITIONS 1-26, BOOK 1, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XXIV. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle ntained by the two sides of one of them greater than the angle contained by the two sides equal to them, of e other; the base of that which has the greater angle, shall be greater than the base of the other. be two triangles, which have the two sides. each to each, namely, equal to and to equal to ; but the greater than the angle each to each, and the angle is equal to the base is equal to the angle And because is equal to in the triangle equal to the angle are equal to the two ; therefore the base therefore the angle is s equal to the angle ; but the angle is greater than the angle herefore the angle he angle is also greater than the angle greater than the angle And because in the triangle ind that the greater angle is subtended by the greater side; therefore the side han Wherefore, if two triangles, &c. the angle is greater than the angle was proved equal to ; therefore is is greater |