The dielectric relaxation in three glass-forming molecular liquids, 1-methylindole (1MID), 5H-5-Methyl-6,7-dihydrocyclopentapyrazine (MDCP), and Quinaldine (QN) is studied focusing on the secondary relaxation and its relation to the structural α-relaxation. All three glass-formers are rigid and more or less planar molecules with related chemical structures but have dipoles of different strengths at different locations. A strong and fast secondary relaxation is detected in the dielectric spectra of 1MID, while no resolved β-relaxation is observed in MDCP and QN. If the observed secondary relaxation in 1MID is identified with the Johari-Goldstein (JG) β-relaxation, then apparently the relation between the α- and β-relaxation frequencies of 1MID is not in accord with the Coupling Model (CM). The possibility of the violation of the prediction in 1MID as due to either the formation of hydrogen-bond induced clusters or the involvement of intramolecular degree of freedom is ruled out. The violation is explained by the secondary relaxation originating from the in-plane rotation of the dipole located on the plane of the rigid molecule, contributing to dielectric loss at higher frequencies and more intense than the JG β-relaxation generated by the out-of-plane rotation. MDCP has smaller dipole moment located in the plane of the molecule; however, presence of the change of curvature of dielectric loss, ε″(f), at some frequency on the high-frequency flank of the α-relaxation reveals the JG β-relaxation in MDCP and which is in accord with the CM prediction. QN has as large an in-plane dipole moment as 1MID, and the absence of the resolved secondary relaxation is explained by the smaller coupling parameter than the latter in the framework of the CM.