RELFE BROTHERS' EUCLID SHEETS Props. 1-26, Book I, are now published in a similar form to this. 164. PROPOSITION XXVI. 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. Let be two triangles which have the angles to and to equal to the angles ; also one side equal to one side. each to each, namely, 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, , one of them must be greater than the other. If possible, let the two sides, is equal to the angle are equal to the two ; therefore the base is equal to the base be greater because each to each; and and the triangle and the other angles to the other angles, each to each, to which the equal sides are is, by the hypothesis, equal is equal to the angle ; wherefore also the angle ; the less angle equal to the greater, ; but the angle is equal to the angle Hence, in the triangles is equal to the angle third angle and the angle to the and the third 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, to , and also the third angle to the third angle to and , one of them must be greater than the other. If possible, let and the other angles to the other angles, each to each, ; but the angle the base and the triangle to which the equal sides are opposite; is equal to the angle terior angle Hence, in the triangles is equal to the angle is equal to its interior and opposite angle that is, is equal to angle is equal to the included angle angle to the third angle ; because is equal to ; therefore the base Wherefore, if two triangles, &c. RELFE BROTHERS, 6, CHARTERHOUSE BUILDINGS, LONDON, E.C. , and the included and the third RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1-26, BOOK 1, 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. not equal to the angle Again, if the angle be less than the base ; but it is not less, therefore the angle is not less than the angle is not equal to the angle ; and it has been shewn, that the angle ; therefore the angle Wherefore, if two triangles, &c. is greater than the angle RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1-26, BOOK I, 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 ontained 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. each to each, and the angle is equal to the base be not greater than ; at the point in the equal to the angle Then, because is the two sides is equal to the angle are equal to the two greater than the angle ; herefore the angle is also greater than the angle ; much more therefore is he angle greater than the angle ; but And because in the triangle and that the greater angle is subtended by the greater side; therefore the side greater than the side than Wherefore, if two triangles, &c. |