To solve this problem, we need to understand the structure of beryllium chloride (BeCl$_2$) in different states.$\\$In the solid state:$\\$• Beryllium chloride forms a polymeric chain structure.$\\$• Each beryllium atom is tetrahedrally coordinated by four chloride ions.$\\$• The structure is a continuous chain due to the bridging chloride ions.$\\$In the vapor phase:$\\$• Beryllium chloride exists as a dimer, $($Be$_2$Cl$_4$).$\\$• This dimeric form is due to the formation of two Be-Cl-Be bridges.$\\$• Each beryllium atom is bonded to two terminal chloride ions and two bridging chloride ions.$\\$Therefore, the correct answer is:$\\$Option 1: Chain and dimer, respectively.$\\$This corresponds to the structure of beryllium chloride in the solid and vapor phases.