PROP. XXVI. THEOR. 27. 1 Eu. straight lines, make the alternate angles Let str. line EF, which falls upon the str. lines AB, CD, make alt. LAEF = alt. ZEFD: then shall AB || CD. For if AB H CD, they will, when produced, meet either towards B, D, or towards A, C. Def. 35. Suppose the former, and let them meet in G, then GEF will form a A; .. ext. L AEF > int. _ EFG; Prop. 15. but L AEF = _ EFG or _EFD, Hyp. which is impossible. ... AB, CD, being produced, do not meet to wards B, D. In like manner it may be proved that they do not meet towards A, C; .. AB || CD; Wherefore if a str. line, &c. Def. 35. PROP. XXVII. THEOR. 28. 1 Eu. 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 shall be parallel to one another. from these equals take BGH, Prop. 26. : AB || CD. Wherefore, if a straight line, &c. PROP. XXVIII. THEOR. 29. I Eu. 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 angle on the same side ; and likewise the two interior angles upon the same side together equal to two right angles. Let the str. line EF fall on the str. lines AB, CD, then alt. L AGH = alt. _ GHD, ext. į EGB= int. Z GHD, |