The degree of entanglement between components of a bipartite system can serve as a useful criterion for studying Quantum Phase Transitions (QPT). \textbf{The von Neumann entropy and generalized concurrence serve as measures, while Schmidt number and Schmidt gap are quantifiers; together, they are key tools for identifying QPT.} Using these \textbf{tools}, we have examined the QPT in the $U(5)\leftrightarrow SU(3)$ region within \textbf{the first version of} the semiclassical approximation \textbf{of the Interacting Boson Model} (IBM-1). Additionally, the Kullback-Leibler divergence \textbf{(KLD)} was employed as a criterion to compare two probability distribution functions to identify the transition region and the critical point. The theoretical results indicated that the Schmidt number and Schmidt gap, the von Neumann entropy, \textbf{generalized concurrence, and} the \textbf{KLD} can characterize the transition behavior and critical point ($X(5)$) in this region. To verify these results in real nuclei, the Dysprosium isotopic chain was selected. Using the suggested entanglement measures, the symmetry limits and the critical point were identified. The results obtained are fully consistent with experimental evidence.
