For focusing on the polar smectic A phase derived from the V-shaped molecules, bent-core molecules with an acute-angled central core (Ar = 1,2-phenylene or 2,3-naphthylene) and a lateral halogen substituent (X = F or Cl) were synthesized. Their mesomorphic properties were investigated by DSC, polarizing microscopy, X-ray and electro-optical measurements. We found that a compound with Ar/X = 2,3-naphthylene/Cl forms the conventional smectic A phase, whereas three compounds with Ar/X = 1,2-phenylene/F, 1,2-phenylene/Cl and 2,3-naphthylene/F form the locally polar smectic A phase (SmAP_R). According to our knowledge this is the first obvious experimental evidence that V-shaped molecules with an acute-subtended angled central core, in cooperation with a lateral halogen substituent, can form the SmAP_R.