## 1 Introduction  
[PERSON] first described a curve that contacts the buccal and lingual cusps of the left and right molars viewed from the coronal section (curve of Wilson) [1]. Such curvature exists due to the lingual inclination of mandibular molars. [PERSON] conducted a study that further supported the lingual inclination of mandibular molars by inserting wires into the root canals of the posterior teeth of skulls [2]. More recently, [PERSON] and [PERSON] studied cone-beam computed tomography (CBCT) images of untreated adults and reported the tendency of buccal inclination of maxillary first molars and lingual inclination in mandibular molars [3]. [PERSON] and [PERSON] studied CBCT images of untreated children and adults and compared their buccolingual inclinations of maxillary and mandibular first molars [4]. They found that both children and adults had lingually inclined mandibular molars and buccally inclined maxillary first molars. However, the adults showed a lesser degree of inclinations, and concluded that first molars, both maxillary and mandibular, become more upright with growth.  
Several studies assessed the association between the molar inclination and vertical facial types. However, these studies were inconclusive. For example, [PERSON] et al. measured the buccolingual inclinations of mandibular molars on dry skulls in different facial types and concluded that mandibular molars were significantly ingally inclined in the hypodivergent group [5]. [PERSON] et al. reported no difference in the mandibular first molar inclinations and facial type [6]. [PERSON] et al. used CBCT images in their study and found that the inclinations of mandibular molars in different facial types were similar [7].  
Other studies investigated the relationship between mandibular alveolar bone inclination and facial types. [PERSON] et al. concluded that the mandibular alveolar bone was significantly more lingually inclined in the hypodivergent group [5]. [PERSON] and [PERSON] studied longitudinal transverse growth of the maxilla and mandible in girls aged 6 to 18 with different vertical facial types [8]. They found that the low mandibular plane angle in hypodivergent children had a wider skeletal maxillary and mandibular transverse dimension at age 6 and the trend continues until age 18. This difference in transverse growth pattern may have an effect on the final mandibular alveolar bone and tooth inclinations. [PERSON] et al. and [PERSON] et al. found that the hyperdivergent group was associated with narrow mandibular arches [9; 10]. By using CBCT scans, [PERSON] et al. evaluated the relationship between mandibular shape and the three-dimensional (3D) skeleton [11]. They found no significant effect of the vertical component of the skull on mandibular morphology.  
To date, there are conflicting results in the literature on the relationship of the inclination of mandibular molars and mandibular alveolar bone, and different vertical facial types. The existing literature only compared either the inclination of mandibular molars or alveolar bone to the vertical skeletal pattern. In addition, some studies used dry skulls to investigate the relationship. In our study, we examined the buccolingual inclination of mandibular first molars and alveolar bones and compared their association with different vertical skeletal patterns. The null hypothesis is that the buccolingual inclination of mandibular first molars and alveolar bones is not related to vertical skeletal patterns.