RELFE BROTHERS EUCLID SHEETS. ' . PROPOSITIONS 1—26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XVI. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. , and Because is equal to to ; the two sides are equal to the two each to each, in the triangles ; and the angle is equal to the angle because they are opposite vertical angles; therefore the base is equal to the base and the triangle to the triangle and the remaining angles of one triangle to the remaining angles of the other, each to each, to which the equal sides are opposite; wherefore the angle is equal to the angle ; but the angle is greater than the angle ; therefore the angle Therefore, if one side of a triangle, &c. RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1--26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XV. If two straight lines cut one another, the vertical, or opposite angles shall be equal. Let the two straight lines cut one another in the point Then the angle shall be equal to the angle and the angle to the Because the straight line makes with at the point the adjacent angles ; these angles are together equal to two right angles. Again, because the straight line makes with at the point the adjacent angles ; these angles also are equal to two right angles; but the angles have been shewn to be equal to two right angles; wherefore the angles are equal to the angles take from each the common angle and the remaining angle is equal to the remaining angle In the same manner it may be demonstrated, that the angle is equal to the angle i Therefore, if two straight lines cut one another, &c. RELFE BROTHERS' EUCLID SHEETS, PROPOSITIONS 1—26, BOOK 1, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XIV. If at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles ; then these two straight lines shall be in one and the same straight line. At the point upon the in the straight line let the two straight lines together equal to two opposite sides of right angles. Then shall be in the same straight line with if possible, let be in the For, if be not in the same straight line with same straight line with it. ; take Then because meets the straight line ; therefore the adjacent angles are equal to two right angles; but the angles are equal to two right angles; therefore the angles are equal to the angles away from these equals the common angle therefore the remaining angle is equal to the remaining angle ; the less angle equal to the greater, which is impossible: therefore is not in the same straight line with And in the same manner it may be demonstrated, that no other can be in the same straight line with it but which therefore is in the same straight line with Wherefore, if at a point, &c. |