If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.
若两条直线从同一点出发，并与另一条直线形成相邻两角且合为两个直角，则这两条直线成一直线。
AB 立在 B 点；BC 向左、BD 向右两边相邻角合为二直角；BE（虚设）与 BC 成一直线，但与 BD 不同，导出矛盾。
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 with any straight line AB, and at the point B on it, let the two straight lines BC, BD not lying on the same side make the adjacent angles ABC, ABD equal to two right angles; I say that BD is in a straight line with CB. For, if BD is not in a straight line with BC, let BE be in a straight line with CB. Then, since the straight line AB stands on the straight line CBE, the angles ABC, ABE are equal to two right angles.
设两条线从同一点在同侧与另一线形成相邻角，且合为两个直角。
[I. 13] But the angles ABC, ABD are also equal to two right angles; therefore the angles CBA, ABE are equal to the angles CBA, ABD. [Post. 4 and C.N. 1] Let the angle CBA be subtracted from each; therefore the remaining angle ABE is equal to the remaining angle ABD, [C.N. 3] the less to the greater: which is impossible.
若这两条线不成一直线，则还存在真正的直线方向与其中一条相邻。
Therefore BE is not in a straight line with CB. Similarly we can prove that neither is any other straight line except BD.
由 euclid-elements/book1-prop-013，这条真正直线形成的相邻角也合为两个直角。
Therefore CB is in a straight line with BD. Therefore etc.
从相等总量中减去公共角，推出两条不同线重合，矛盾；故两线成一直线。