Euclid's Sixth Proposition in the Poincaré Disk - http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI6.htmlIf in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
I say that the side AB also equals the side AC.
If AB does not equal AC, then one of them is greater.
Let AB be greater. Cut off DB from AB the greater equal to AC the less, and join DC.
Since DB equals AC, and BC is common, therefore the two sides DB and BC equal the two sides AC and CB respectively, and the angle DBC equals the angle ACB. Therefore the base DC equals the base AB, and the triangle DBC equals the triangle ACB, the less equals the greater, which is absurd. Therefore AB is not unequal to AC, it therefore equals it.
Therefore if in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.