If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.
一条直线立在另一条直线上时，所成的相邻两角或者都是直角，或者合起来等于两个直角。
直线 CD 上点 B；AB 从 B 斜向上；BE 是 B 处对 CD 的垂线（辅助）。
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 any straight line AB set up on the straight line CD make the angles CBA, ABD; I say that the angles CBA, ABD are either two right angles or equal to two right angles. Now, if the angle CBA is equal to the angle ABD, they are two right angles. [Def. 10] But, if not, let BE be drawn from the point B at right angles to CD; [I. 11] therefore the angles CBE, EBD are two right angles.
一条直线立在另一条直线上，若相邻两角相等，则按定义它们都是直角。
Then, since the angle CBE is equal to the two angles CBA, ABE, let the angle EBD be added to each; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. [C. N. 2] Again, since the angle DBA is equal to the two angles DBE, EBA, let the angle ABC be added to each; therefore the angles DBA.
若不相等，从立足点作一条垂线，得到两个直角。
ABC are equal to the three angles DBE, EBA, ABC. [C. N. 2] But the angles CBE, EBD were also proved equal to the same three angles; and things which are equal to the same thing are also equal to one another; [C. N. 1] therefore the angles CBE, EBD are also equal to the angles DBA, ABC.
把同一侧角分解并用公理 2、3 加减相等角。
But the angles CBE, EBD are two right angles; therefore the angles DBA, ABC are also equal to two right angles. Therefore etc.
可得原来的两个相邻角合起来等于两个直角。