Abstract In [Comput.~Math.~Appl. 41 (2001), 135--147],
A.A. Ungar employs the Möbius gyrovector spaces for the
introduction of the hyperbolic trigonometry. This A.A. Ungar's
work, plays a major role in translating some theorems in Euclidean
geometry to corresponding theorems in hyperbolic geometry. In this
paper we present (i)~the hyperbolic Breusch's lemma, (ii)~ the
hyperbolic Urquhart's theorem, and (iii)~ the hyperbolic
Steiner-Lehmus theorem in the Poincaré ball model of
hyperbolic geometry by employing results from A.A. Ungar's work.
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