Proof: By Euclid
(related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
 By Prop 1.32, the sum of all angles in a plane triangle is $180^\circ.$
 A triangulation of a quadrilateral gives two triangles, with the sum of $2\cdot 180^\circ=4\cdot 90^\circ$ right angles.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Calahan
 @Casey
References
Adapted from CC BYSA 3.0 Sources:
 Callahan, Daniel: "Euclidâ€™s 'Elements' Redux" 2014
Adapted from (Public Domain)
 Casey, John: "The First Six Books of the Elements of Euclid"