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 because they are opposite vertical angles; therefore the base is equal to the angle is equal to the base 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 ; but the angle is greater than the angle is equal to the angle ; therefore the angle is greater than the angle In the same manner, if the side may be demonstrated that the angle angle Therefore, if one side of a triangle, &c. be bisected, and be produced to ; it that is, the angle is greater than the 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. at the point , the adjacent ; these angles are together equal to two right angles. Again, because the straight line angles makes with ; 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 ; take away from each the common are equal to the angles is equal to the remaining angle angle and the remaining angle the same manner it may be demonstrated, that the angle In is equal to the angle Therefore, if two straight lines cut one another, &c. PROPOSITIONS 1—26, BOOK I, 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 opposite sides of in the straight line let the two straight lines upon the together equal to two therefore the adjacent angles are equal to are equal to two right angles; but the angles two right angles; therefore the angles ; take away from these equals the common angle angle is equal to the remaining angle which is impossible: therefore are equal to the angles therefore the remaining ; the less angle equal to the greater, 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. |