RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION VIII. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other. be applied to then because wherefore shall be equal to the angle For, if the triangle is equal to so that the point on ; shall coincide with the point ; coinciding with and shall coincide with ; for, same base, and upon the same side of it, there can be two triangles which have their sides which are terminated in one extremity of the base, equal to one another, and likewise those sides which are terminated in the other extremity; but this is impossible. Therefore, if two triangles have two sides, &c. RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION VII. Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base, equal to one another, and likewise those which are terminated in the other extremity. and upon the same side of it, let there be two triangles Join First. When the vertex of each of the triangles is without the other triangle. Because is equal to angle ; but the angle is greater than the side is equal to ; but the angle both equal to, and greater than the angle in the triangle therefore the angle greater than ; therefore also the angle is equal to the angle hence the angle is to and therefore the angles to one another; but the angle is greater than the angle Again, because to and join Then because is equal to upon the other side of the base in the are equal is greater than the angle ; much more then is the angle in the triangle is both equal to, and greater than the angle ; which is impossible. Thirdly. The case in which the vertex of one triangle is upon a side of the other, needs no demonstration. Therefore, upon the same base and on the same side of it, &c. |