<p>The characteristic quantities and their relationships in the context of quantum teleportation (QT) of an arbitrary qubit state (AQS), where the initial state (IS) of the quantum channel (QC) is a Bell-like state (BLS), have been investigated. The qubits of the QC are independently affected by an amplitude-damping noise environment (ADNE). We have derived analytical expressions for the complementary relationship between the purity measure (PM) of the channel and the entanglement measures (EMs) between each qubit of the channel and the environmental qubit (EQ). Additionally, we have established analytical expressions describing the dependence of the average quantum fidelity (AQF) of the protocol on the channel PM, the channel EM, the bipartite EMs involving channel and EQs, as well as the multipartite entanglement between the channel qubits (CQs) and the EQs. Based on these relationships, we have revealed the physical mechanism underlying the enhancement or non-enhancement of the average fidelity (AF) when the remaining CQ is affected by amplitude-damping noise (ADN). Furthermore, we have analyzed the optimization process of the AF for two protocols in which the IS of the QC is chosen as a BLS.</p>