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Solution

$x+x=x$ is Idempotent Law, so $A-III$.

$x+0=x$ is Identity Law, so $B-I$.

$x+1=1$ is Domination Law, so $C-IV$.

$x+xy=x$ is Absorption Law, so $D-II$.

Final matching is $A-III,\ B-I,\ C-IV,\ D-II$.