Euclid's Elements Book I Proposition 16
In any triangle, if one of the sides is produced, then the exterior angle is greater than either of the interior and opposite angles.
In other words:
Begin with triangle ABC.
Extend line BC to point D.
The angle ACD is larger than either ABC or CAB.
Steps and Proof
Proof
1) Draw triangle ABC
2) Extend line segment BC beyond point C to point D
3) The exterior angle ACD is greater than either of the interior and opposite angles CBA and BAC
Steps of the construction
1) Bisect line AC at point E (I, 10).
2) Create line segment BE.
3) Extend line BE to point F; where BE is equal to EF (I, 3).
4) Create line CF (Post 1).
5) Extend line AC to point G (Post 2).
6) Since AE equals EC, and BE equals EF, the two sides AE and EB equal the two sides CE and EF.
7) Angles AEB and CEF are vertical angles and are equal (I, 15).
8) Base AB equals base FC, triangle ABE equals the triangle CFE, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. (I, 4)
9) Therefore the angle BAE equals the angle ECF.
10) Angle ECD is greater than Angle ECF (C.N. 5).
11) Therefore, angle ACD is greater than angle BAE.
12) Also, if BC is bisected, the angle BCG, which is equal to angle ACD (I, 15) can be proved greater than angle ABC as well.
13) Therefore, the exterior angle ACD is greater than either of the interior and opposite angles CBA and BAC.