Proof: By Euclid
(related to Corollary: Angles of Right Triangle)
 By Prop 1.32, the sum of all angles in a plane triangle is $\alpha+\beta+\gamma=180^\circ.$
 Without loss of generality, let $\alpha+\gamma=\beta.$
 Then $2\beta=180^\circ,$ and $\beta=90$ is a right angle.
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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"