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 $\gamma_1=\gamma_2.$
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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"