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.