PROPOSITION XXVII. THEOREM. If a straight line, falling upon two other straight lines, make the alternate angles equal to one another; these two straight lines must be parallel. Let the st. line EF, falling on the st. lines AB, CD, make the alternate 4s AGH, GHD equal. Then must AB be || to CD. For if not, AB and CD will meet, if produced, either towards B, D, or towards A, C. But Let them be produced and meet towards B, D in K. Then GHK is a A; and .. AGH is greater than ▲ GHD. which is impossible. LAGH= LGHD, ·. AB, CD do not meet when produced towards B, D. I. 16. Нур. In like manner it may be shewn that they do not meet when produced towards A, C. .. AB and CD are parallel. Def. 26. Q.E. D. PROPOSITION XXVIII. THEOREM, If a straight line, falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line, or make the interior angles upon the same side together equal to two right angles; the two straight lines are parallel to one another. Let the st. line EF, falling on st. lines AB, CD, make I. LEGB = corresponding ▲ GHD, or II. ▲ 8 BGH, GHD together = two rt. 48. Then, in either case, AB must be || to CD. and 48 BGH, AGH together = two rt. 4s, I. 27. Hyp. I. 13. .. 48 BGH, AGH together: = 48 BGH, GHD together; .. LAGH= L GHD ; .. AB is to CD. I. 27. Q. F. D. NOTE V. On the Sixth Postulate. We explained in Note Iv., page 32, that Euclid's Sixth Postulate is the converse of the 17th Proposition. In the place of this Postulate many modern writers on Geometry propose, as more evident to the senses, the following Postulate: "Two straight lines which cut one another cannot BOTH be parallel to the same straight line." If this be assumed, we can prove Post. 6, as a Theorem, thus: Let the line EF falling on the lines AB, CD make the LS BGH, GHD together less than two rt. 4 s. Then must AB, CD meet when produced towards B, D. For if not, suppose AB and CD to be parallel. I. 13. and 48 GHD, BGH are together less than two rt. 4s, Make .. LAGH is greater than GHD. MGH= L GHD, and produce MG to N. .. MN is to CD. I. 27. Thus two lines MN, AB which cut one another are both parallel to CD, which is impossible. .. AB and CD are not parallel. It is also clear that they meet towards B, D, because GB lies between GN and HD. Q. E. D. PROPOSITION XXIX. THEOREM. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles. Let the st. line EF fall on the parallel st. lines AB, CD. III. LS BGH, GHD together two rt. Ls. = Then 48 AGH, BGH are together greater than ≤ 8 GHD, BGH together. Now 48 AGH, BGH together two rt. <s; :. 4ș GHD, BGH are together less than two rt. :. AB and CD will meet if produced towards B, D. But they cannot meet, they are parallel; :. LAGH is not greater than ▲ GHD. Similarly it may be shewn that LAGH is not less than 4 GHD; I. 13. 2s; Post. 6. III. . GHD has been proved = ▲ EGB, .. adding to each BGH, 48 BGH, GHD together= 48 BGH, EGB together. But 48 BGH, EGB together=two rt. 48; I. 13. ... 48 BGH, GHD together=two rt. 4 s. Q. E. D. EXERCISES. 1. If through a point, equidistant from two parallel straight lines, two straight lines be drawn cutting the parallel straight lines; they will intercept equal portions of those lines. 2. If a straight line be drawn, bisecting one of the angles of a triangle, to meet the opposite side; the straight lines drawn from the point of section, parallel to the other sides and terminated by those sides, will be equal. 3. If any straight line joining two parallel straight lines be bisected, any other straight line, drawn through the point of bisection to meet the two lines, will be bisected in that point. 8. E. |