Given $Γ$ is semantically inconsistent, need to prove: for all propositions $θ$, we have $Γ⊨θ$.
"$Γ$ is semantically inconsistent", then there is no truth assignment $A$ such that $A(Γ)=1$.
But the definition of entail states:
A set of sentences $Δ$ logically entails a sentence $φ$ (written $Δ⊨φ$ ) if and only if every truth assignment that satisfies $Δ$ also satisfies $φ$.
My question is: Since there is not truth assignment $A$ satisfies $A(Γ)=1$, how can we find a truth assignment satisfies $φ$?
You don't need to find a truth-assignment that satisfies $\varphi$. What you need to prove is that if a truth-assignment satisfies $\Gamma$, then it satisfies $\varphi$. And to prove that, it suffices to point out that there is no truth-assignment that satisfies $\Gamma$: with the antecedent part false, the whole conditional is (vaciously) true.