In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.
三角形的一边延长时，外角等于两个内对角之和，三内角合为两个直角。
三角形 ABC，BC 延长到 D；过 C 作 CE 平行于 AB；外角 ACD 被分为 ACE 与 ECD，分别等于 BAC 与 ABC。
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, and let one side of it BC be produced to D; I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, ABC, and the three interior angles of the triangle ABC, BCA, CAB are equal to two right angles. For let CE be drawn through the point C parallel to the straight line AB.
过三角形顶点作一条直线，平行于对边（euclid-elements/book1-prop-031）。
[I. 31] Then, since AB is parallel to CE, and AC has fallen upon them, the alternate angles BAC, ACE are equal to one another. [I. 29] Again, since AB is parallel to CE, and the straight line BD has fallen upon them, the exterior angle ECD is equal to the interior and opposite angle ABC.
由 euclid-elements/book1-prop-029，外角被分成两个角，分别等于两个内对角。
[I. 29] But the angle ACE was also proved equal to the angle BAC; therefore the whole angle ACD is equal to the two interior and opposite angles BAC, ABC. Let the angle ACB be added to each; therefore the angles ACD, ACB are equal to the three angles ABC, BCA, CAB.
所以外角等于两个内对角之和。
But the angles ACD, ACB are equal to two right angles; [I. 13] therefore the angles ABC, BCA, CAB are also equal to two right angles. Therefore etc.
外角与相邻内角合为两个直角，故三内角合为两个直角。