If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.
一条直线落在两条直线上，若同位角相等，或同旁内角合为两个直角，则这两条直线平行。
直线 AB 与 CD 被横截线 EF 截于 G、H；同位角 EGB=GHD 或同旁内角合二直角，皆推出 AB∥CD。
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For let the straight line EF falling on the two straight lines AB, CD make the exterior angle EGB equal to the interior and opposite angle GHD, or the interior angles on the same side, namely BGH, GHD, equal to two right angles; I say that AB is parallel to CD. For, since the angle EGB is equal to the angle GHD, while the angle EGB is equal to the angle AGH, [I. 15] the angle AGH is also equal to the angle GHD; and they are alternate; therefore AB is parallel to CD.
同位角相等时，可先由相邻角合二直角把它转成内错角相等。
[I. 27] Again, since the angles BGH, GHD are equal to two right angles, and the angles AGH, BGH are also equal to two right angles, [I. 13] the angles AGH, BGH are equal to the angles BGH, GHD.
同旁内角合二直角时，也可通过相邻角相等等价转成内错角相等。
Let the angle BGH be subtracted from each; therefore the remaining angle AGH is equal to the remaining angle GHD; and they are alternate; therefore AB is parallel to CD.
一旦内错角相等，就由 euclid-elements/book1-prop-027 得平行。
[I. 27] Therefore etc.
故两种角条件都推出两直线平行。