interior and exterior angle
This figure can explain the relation between an exterior angle of a triangle and the sum of the interior opposite angles, based on properties of parallel lines.
In The Elements of Euclid, this theorem of exterior angle precedes the theorem about angle sum of a triangle.
The following explanation is extracted from a geometry textbook of the Chiu Chang Publisher (with Chinese and English version available online).
Before the proof, an analysis is given to explain the rationale behind adding an auxiliary line. This analysis also suggests how the proof is related to the intuitive idea of moving and rotating the angles, which is also commonly experienced or demonstrated with paper cutting. (See also relevant activities in section 2 or 3 of this collection.)
The proof basically explains both the interior angle sum property and the exterior angle property. The latter is mentioned as a corollary.
The following silent video shows this construction.