In any triangle two angles taken together in any manner are less than two right angles.
任一三角形中，任意两角合起来小于两个直角。
三角形 ABC，BC 延长到 D。外角 ACD 大于内对角 ABC，因此 ABC 与 ACB 合起来小于二直角。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let ABC be a triangle; I say that two angles of the triangle ABC taken together in any manner are less than two right angles. For let BC be produced to D.
延长三角形的一边，形成外角。
[Post. 2] Then, since the angle ACD is an exterior angle of the triangle ABC, it is greater than the interior and opposite angle ABC. [I. 16] Let the angle ACB be added to each; therefore the angles ACD, ACB are greater than the angles ABC, BCA.
由 euclid-elements/book1-prop-016，外角大于任一个内对角。
But the angles ACD, ACB are equal to two right angles. [I. 13] Therefore the angles ABC, BCA are less than two right angles.
外角与相邻内角合为两个直角（euclid-elements/book1-prop-013）。
Similarly we can prove that the angles BAC, ACB are also less than two right angles, and so are the angles CAB, ABC as well. Therefore etc.
把“大于”的外角换成内对角，可知任意两内角之和小于两个直角。