RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION VI. If two angles of a triangle be equal to each other; the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another. PROPOSITIONS 1—26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS PROPOSITION V. The angles at the base of an isosceles triangle are equal to each other; and if the equal sides be produced, the angles on the other side of the base shall be equal. also the remaining angles of the one are equal to the remaining angles of the other, each to each, to which the equal sides are opposite; viz., the angle ; to the angle and the angle to the angle And because the whole proved to be equal to ; hence, because the two sides is equal to the whole of which the parts has been has been proved to be equal to the angle also the base is common to the two triangles ; wherefore these triangles are equal, and their And, since it has been demonstrated, that the whole angle is the parts of which, the angles equal to the remaining angle is equal to the whole are also equal; therefore the remaining angle which are the angles at the base of the triangle is ; and it has also been proved, that the angle is equal to the angle which are the angles upon the other side of the base. Therefore the angles at the base, &c. PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION IV. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by those sides equal to each other; they shall likewise have their bases or third sides equal, and the two triangles shall be equal, and their other angles shall be equal, each to each, viz., those to which the equal sides are opposite. be two triangles, which have the two sides each to each, viz., to and to equal to the and the included Let two sides angle equal to the included angle Then shall the base be equal to the base ; and the triangle to the triangle ; and the other angles to which the equal sides are opposite shall be equal, shall fall on because the angle also the point shall coincide with the point do not coincide with the base enclose a space, which is impossible. Therefore the base triangle is equal to the angle because the straight line therefore is equal to ; but the shall coincide if the base and would does coincide with coincides with the whole triangle remaining angles of one triangle coincide with the equal to them, viz., the angle to the angle and is equal to it; also the remaining angles of the other, and are Therefore, if two triangles have two sides of the one equal to two sides, &c. to |