If two straight lines cut one another, they make the vertical angles equal to one another.
两条直线相交时，对顶角彼此相等。
两直线 AB、CD 交于 E 点；A、B 在两端，C、D 在另两端。
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 lines AB, CD cut one another at the point E; I say that the angle AEC is equal to the angle DEB, and the angle CEB to the angle AED. For, since the straight line AE stands on the straight line CD, making the angles CEA, AED, the angles CEA, AED are equal to two right angles [I. 13] Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB, the angles AED, DEB are equal to two right angles. [I. 13] But the angles CEA, AED were also proved equal to two right angles; therefore the angles CEA, AED are equal to the angles AED DEB.
两条直线相交时，每一对相邻角都由 euclid-elements/book1-prop-013 合为两个直角。
[Post. 4 and C. N. 1] Let the angle AED be subtracted from each; therefore the remaining angle CEA is equal to the remaining angle BED. [C. N. 3] Similarly it can be proved that the angles CEB, DEA are also equal.
取一对对顶角，它们分别是从两个二直角总量中减去同一个相邻角所得。
Therefore etc. Q. E.
由公理 3，余角相等。
D. Porism.
另一对对顶角同理相等。