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. RELFE BROTHERS' EUCLID SHEETS. 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. sides is equal to Let be an isosceles triangle of which the side and let the equal the greater, cut are equal to the two 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 ; hence, because the two sides is equal to the whole , of which the parts has been are equal; therefore the remainder proved to be equal to each to each; and the angle ; wherefore these triangles are equal, and their remaining angles, each to each, to which the equal sides are opposite; therefore the angle 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. RELFE BROTHERS' EUCLID SHEETS. 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 triangle ; and the other angles to which the equal sides are opposite shall be equal, each to each, viz., the angle to the angle and the angle to the to the shall coincide with the point, because was shewn to coincide with the point ; wherefore the base shall coincide ; then the point may the straight line therefore is equal to ; but the coinciding with and with the two straight lines and would and is equal to it; and the whole and is equal to it; also the e triangle coincide with the remaining angles of the other, and are and the angle les have two sides of the one equal to two sides, &c. to |