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Let $(L,\wedge ,\vee )$ be a finite lattice with a least element 0. $𝔸G\left(L\right)$ is an annihilating-ideal graph of $L$ in which the vertex set is the set of all nontrivial ideals of $L$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I\wedge J=0$. We completely characterize all finite lattices $L$ whose line graph associated to an annihilating-ideal graph, denoted by $𝔏\left(𝔸G\right(L\left)\right)$, is a planar or projective graph.