There is likely a misprint on pages 31,32 and 33 of the book "Geometric Theory of Functions of a Complex Variable" by G.M. Goluzin, given below (see Lemma 1 and its proof ):
This is because in the proof given by Goluzin, there must be some relationship between the points $a$ and $b$ and the Jordan arcs $\lambda_n$, or else we should not always be able to find a sector in which there are infinitely many Jordan arcs with subarcs having endpoints on the boundary radii of the sector, which is needed to carry forward Goluzin's proof of Lemma 1 (attributed to Koebe).
I am looking for what is likely the correct printing of this Lemma due to Koebe, which I have been unable to find in the bibliography of Goluzin's book. This may be inferable from just reading Goluzin's proof, but I do not know of a way to reasonably correct the printing of Lemma 1 as it stands (for one, it seems the Jordan arcs cannot have as a limiting value an interior point of $\mathbb{D}$, or else the theorem should not hold).