Proof: By Euclid
(related to Corollary: Similar Triangles)
- By Prop 1.32, the sum of all angles in a plane triangle is $180^\circ.$
- This holds for any two triangles in the plane.
- By hypothesis, the first triangle has sum of angles $\alpha+\beta+\gamma_1=180^\circ$ and the second triangle the sum $\alpha+\beta+\gamma_2=180^\circ.$
- Therefore $\alpha+\beta+\gamma_1=180^\circ=\alpha+\beta+\gamma_2,$ or $\gamma_1=\gamma_2.$
Thank you to the contributors under CC BY-SA 4.0!
Adapted from CC BY-SA 3.0 Sources:
- Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
- Casey, John: "The First Six Books of the Elements of Euclid"