Bibenzoate copolyesters are based on (4,4′-biphenyl dicarboxylic acid-co-3,4′-biphenyl dicarboxylic acid) as the diacid component, and on an alicyclic diol compound such as 1,4-cyclohexanedimethanol as a portion of the diol component. Copolyesters are based on 4,4′-biphenyl dicarboxylic acid, and/or 3,4′-biphenyl dicarboxylic acid as the diacid component and may include a multifunctional acid. Copolymers may optionally base an essentially amorphous morphology, high glass transition temperature, high elongation at break, and/or high melting temperature. A method to make the copolymers controls the characteristics of the copolyester selected from one or a combination of amorphous morphology or degree of crystallinity, Tg, Tm, tensile modulus, flexural modulus, elongation at break, and so on, by selecting the proportions of the 4,4′-biphenyl dicarboxylic acid or ester producing equivalent thereof, 3,4′-biphenyl dicarboxylic acid or ester producing equivalent thereof, and/or the proportion of the 1,4-cyclohexanedimethanol in the diol component.