Certain ergodic, piecewise M\"{o}bius self-mappings of the unit interval, similar to the classical Gauss or R\'enyi maps, give rise to natural sequences of convergents $p_n/q_n$ for every associated ``irrational'' number $x$. Here we study the metric theory of the approximation sequences $\theta_n=|q_n| |q_n x-p_n|$. Following Jager we describe the distribution of pairs $( \theta_n,\theta_{n+1})$ in a plane domain by deriving their distribution function. As a consequence we get a generalization of the theorem of Bosma, Jager and Wiedijk, referred to as the Lenstra Conjecture, which describes the distribution of the $\theta_n$.