For any triangle, when one of the sides is produced, the external angle is greater than each of the internal and opposite angles. * Let $ABC$ be a triangle, and let one of its sides $BC$ have been produced to $D$. * I say that the external angle $ACD$ is greater than each of the internal and opposite angles, $CBA$ and $BAC$.
Construct \(\triangle{ABC}\) and extend any of its sides, e.g. \(\overline{BC}\), to the segment \(\overline{CD}\). Then the exterior angle \(\gamma=\angle{DCA}\) is greater than either of the interior non-adjacent angles \(\alpha=\angle{CBA}\) and \(\beta=\angle{BAC}\).
Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9