A B and C are two straight lines met by the E A B C D F straight line EF, and the 28 BEF and EFD are together equal to two right 28. It has to be proved that the right lines A B and CD are parallel. Proof. Because the straight line FE meets the straight line AB, the adjacent s AEF and BEF are together equal to two rights. (Prop. XIII.) But the s BEF and EFD are also together equal to two rights. Therefore the sum of the s AEF and BEF is equal to the sum of the s BEF and EF D. If from each of these equal sums we take away the BEF, the remainders are equal. That is to say, the AEF is equal to the EFD. But A EF and EFD are alternates. Therefore, as these alternate s are equal, the lines A B and CD will never meet when produced; or in other words are parallel. (Prop. XXVII.) Thus far there is no difficulty whatever with respect to parallel lines. The proof of the next proposition, however, requires that we should assume the truth of the 11th axiom. Now this axiom, though no one ever doubted its truth, is by no means so obviously true as the other axioms. In fact there are several propositions of which a detailed proof is given in geometry, the truth of which is more readily perceived than that of the 11th axiom. Numerous attempts have, in consequence, been made to supersede the necessity for the use of this axiom; but entirely without success, as some axiom or other is always assumed which is quite as difficult as the 11th. The difficulty, of course, arises from the fact that the definition of parallel lines is of a negative character, and therefore, without some positive axiom, would not enable us to found any inferences upon it. PROPOSITION XXIX. If two straight lines are parallel and a third straight line intersects them, then the alternate angles so formed will be equal, the exterior angle will be equal to the interior and opposite angle on the same side of the intersecting line, and the two interior angles on the same side of the intersecting lino will together be equal to two right angles. For the proof of this proposition we must know,— 1. That if two straight lines meet a third in such a way that the two interiors on the same side of it are together less than two right 28, those straight lines will meet if produced. (Ax. XI.) 2. That magnitudes that are equal to the same are equal to one another. (Ax. I.) 3. That if equals or the same be taken from equals, the remainders are equal. (Ax. III.) 4. That if one straight line meets another, the adjacents so formed are together equal to two rights. (Prop. XIII.) 5. That if two straight lines intersect one another, the vertically opposites are equal. (Prop. XV.) The 29th proposition in reality consists of three distinct propositions, classed together under one heading or enunciation, and corresponding respectively to the 27th and the two parts of the 28th proposition, of which they are the converse. As all three cases of the 29th proposition depend upon the application of the 11th axiom, it will be most convenient to take the third case first, as in it the axiom in question is applied most directly. 1. If two straight lines are parallel, and a third meets them, the two interior angles on the same side of the intersecting line are together equal to two right angles. E The two lines A B and CD are parallel (i.e., however far they may be produced, they will never meet). The line EH intersects them. We have to [rove that the s BFL and FLD are together equal to two rights. F A L C H B -D The proof of this is of the indirect kind. If the s BFL and FLD are not together equal to two rights, they must be either less than two rights, or greater than two right ▲s. If they were less than two rights, then the lines FB and LD would meet if produced (according to the 11th axiom). But this is impossible, for we know, to start with, that the lines A B and CD are parallel (i.e., can never meet). Consequently the supposition that would lead to this impossibility must be itself impossible. That is to say, it is impossible that the s BFL and FLD should be together less than two rights. Again-suppose the s BFL and FLD were together greater than two rights. Since the s BFL and LFA are together equal to two rights (according to Prop. XIII.), and the /s FLD and FLC are also equal to two rights, the four 8 BFL, LF A, FLD, and FLC are together equal to four rights. Therefore, it would follow that on taking away from the sum of these fours the two s BFL and FLD, if the latter were together greater than two rights, the remainings AFL and FLC would be together less than two rights. But if the two straight lines A F and CL meeting a third, FL, made the interior s AFL and FLC together less than two rights, the two straight lines FA and LC would meet if produced. (Ax. XI.) But this is impossible, because we know, to start with, that FA and LC are parallel. Consequently, it is impossible that the two s AFL and FLC can be together less than two right Zs, and therefore it is impossible that the two s BFL and FL D should be together greater than two rights. Hence, as the s BFL and FLD can be neither less than two right s nor greater than two rights, they must together be equal to two rights. Moreover, since BFL and FLD are together equal to two rights, if they be taken away from the four Zs BFL, LFA, FLD, and FLC, the remaining Zs LFA and FLC will be together equal to two rights. 2. If two straight lines are parallel and a third straight line intersects them, the alternate angles so formed will be equal. Let A B and C D be parallel, and let E H intersect BFL and FLD are together equal to two rights. Also, since the straight line LF meets the straight line AB, the adjacent s AFL and BFL are together equal to two rights. (Prop. XIII.) Therefore the sum of the 8 BFL and FLD is equal to the sum of the ▲s AFL and L F B. (Ax. I.) Take away the common, BF L, and the remainders, the s AFL and FLD will be equal to one another. Also, it was proved in the last case that the two interiors AFL and FLC are together equal to two rights. But the s AFL and BFL are also together equal to two rights. Therefore the s AFL and FLC are together equal to the sum of the two s AFL and LF B. Take away the common, AF L, and the remainders, namely, the s BFL and F L C, will be equal. 3. If two straight lines are parallel and a third straight line intersects them, each of the exterior angles is equal to the interior and opposite angle on the same side of the intersecting line. E Let A B and CD be parallel, and let E H intersect them in the points F and L. We have to prove that the ZEF B is equal to the FLD, and the F A B the FLC. с D L H EFA to Proof. Since the straight lines A B and EH intersect one another at the point F, the vertically opposite s E F B and A F L are equal. (Prop. XV.) But it was proved in the last case that the AFL is equal to the FLD. Therefore the EFB is also equal to the FLD. Again, the vertically opposite s EFA and B FL are equal. (Prop. XV.) |