Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/43

 PROPOSITION 14. THEOREM. If, at a point in a straight line, two other straight lines, on the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. At the point B in the straight line AB, let the two straight lines BC, BD, on the opposite sides of AB, make the adjacent angles ABC, ABD together equal to two right angles: BD shall be in the same straight line with CB. For if BD be not in a the same straight line with CB, let BE be in the same straight line with it. Then because the straight line AB makes with the straight line CBE, on one side of it, the angles ABC, ABE, these angles are together equal to two right angles. But the angles ABC, ABD are also together equal to two right angles. Therefore the angles ABC, ABE are equal to the angles ABC, ABD. From each of these equals take away the common angle ABC, and the remaining angle ABE is equal to the remaining angle ABD, the less to the greater; which is impossible. Therefore BE is not in the same straight line with CB. And in the same manner it may be shewn that no other can be in the same straight line with it but BD; therefore BD is in the same straight line with CB. Wherefore, if at a point &c. PROPOSITION 15. THEOREM. If two straight lines cut one another, the vertical, or opposite, angles shall be equal. 2—2