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This paper studies flag-transitive point-primitive non-symmetric$2$-($v,k,2$) designs. We prove that if $\mathcal{D}$ is a non-trivialnon-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitiveautomorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ isa $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.

Flag-transitive point-primitive non-symmetric 2-(v,k,2) designs with alternating socle