= Solution
If two <hyperbolic lines> intersect, move their intersection to the centre of the <Poincaré disk model>. Lines through the centre are diameters. Reflection in a diameter making Euclidean angle $\alpha$ with the real axis is $z\mapsto e^{2i\alpha}\overline z$. The composition of two such reflections is the rotation $z\mapsto e^{2i(\alpha-\beta)}z$. The <Poincaré disk model> is conformal, so the Euclidean angle difference is the hyperbolic intersection angle $\theta$, up to sign and replacing it by $\pi-\theta$, neither of which changes rationality of $\theta/\pi$.
The rotation has finite order exactly when $m\theta\in\pi\mathbb Z$ for some positive integer $m$, equivalently $\theta/\pi\in\mathbb Q$. If $\theta/\pi=r/s$ in lowest terms, its order is $s$. Conversely, the nonintersecting cases in part(a) have infinite order. Therefore
$$
\boxed{R_1R_2\text{ has finite order precisely when the lines meet at a rational multiple of }\pi.}
$$
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