this paper, we introduce a notion of approximate -biprojectivity for the enveloping dual Banach algebras. We also find some relations between approximate Connes amenability, approximate biprojectivity, left -contractibility and approximate -biprojectivity. Moreover, we propose a criterion to show that the enveloping dual Banach algebras associated to triangular Banach algebras are not approximately -biprojective. Finally, we present some examples of the enveloping dual Banach algebras associated to and (equipped with a new multiplication) and also study their approximate -biprojectivity, where is a unital weakly cancellative
