In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.
任一三角形的一边延长后，外角大于任一个内对角。
三角形 ABC，BC 向 D 延长；E 是 AC 中点；BE 延长到 F 使 EF=BE；连 CF；AC 向 G 延长。外角 ACD 大于内对角。
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Let ABC be a triangle, and let one side of it BC be produced to D; I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA, BAC. Let AC be bisected at E [I. 10], and let BE be joined and produced in a straight line to F; let EF be made equal to BE[I. 3], let FC be joined [Post. 1], and let AC be drawn through to G [Post. 2].
延长三角形的一边，并取延长段上的辅助点。
Then, since AE is equal to EC, and BE to EF, the two sides AE, EB are equal to the two sides CE, EF respectively; and the angle AEB is equal to the angle FEC, for they are vertical angles. [I. 15] Therefore the base AB is equal to the base FC, and the triangle ABE is equal to the triangle CFE, and the remaining angles are equal to the remaining angles respectively, namely those which the equal sides subtend; [I. 4] therefore the angle BAE is equal to the angle ECF.
平分相邻边或作相等辅助线，构造两个可比较的小三角形。
But the angle ECD is greater than the angle ECF; [C. N. 5] therefore the angle ACD is greater than the angle BAE. Similarly also, if BC be bisected, the angle BCG, that is, the angle ACD [I. 15], can be proved greater than the angle ABC as well.
由 euclid-elements/book1-prop-004 得到一个角等于内对角，而外角包含这个角。
Therefore etc.
整体大于部分（公理 5），所以外角大于任一内对角。