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We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X$, the Lipschitz algebras ${\mathrm{Lip}}_{\alpha }\left(X\right)$ and ${\mathrm{lip}}_{\alpha }\left(X\right)$ are approximately biflat if and only if $X$ is finite, provided that $0<\alpha <1$. We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.