A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles.
一条直线落在两条平行直线上，则内错角相等，同位角相等，同旁内角合为两个直角。
AB 与 CD 是平行直线；EF 是横截线，交 AB 于 G，交 CD 于 H。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For let the straight line EF fall on the parallel straight lines AB, CD; I say that it makes the alternate angles AGH, GHD equal, the exterior angle EGB equal to the interior and opposite angle GHD, and the interior angles on the same side, namely BGH, GHD, equal to two right angles. For, if the angle AGH is unequal to the angle GHD, one of them is greater. Let the angle AGH be greater.
一条直线落在两条平行直线上。
Let the angle BGH be added to each; therefore the angles AGH, BGH are greater than the angles BGH, GHD. But the angles AGH, BGH are equal to two right angles; [I. 13] therefore the angles BGH, GHD are less than two right angles. But straight lines produced indefinitely from angles less than two right angles meet; [Post. 5] therefore AB, CD, if produced indefinitely, will meet; but they do not meet, because they are by hypothesis parallel.
若内错角不相等，则某一侧同旁内角和会小于两个直角。
Therefore the angle AGH is not unequal to the angle GHD, and is therefore equal to it. Again, the angle AGH is equal to the angle EGB; [I. 15] therefore the angle EGB is also equal to the angle GHD. [C.N. 1] Let the angle BGH be added to each; therefore the angles EGB, BGH are equal to the angles BGH, GHD.
由公设 5，两平行线会在该侧相交，矛盾。
[C.N. 2] But the angles EGB, BGH are equal to two right angles; [I. 13] therefore the angles BGH, GHD are also equal to two right angles. Therefore etc.
所以内错角相等；同位角相等和同旁内角合二直角再由相邻角关系推出。