  In this note we introduce the notion of $T^*-$extension $T^*{\mathfrak g}$ of
a Lie superalgebra ${\mathfrak g}$, i.e. an extension of ${\mathfrak g}$ by its
dual space ${\mathfrak g}^*$. The natural pairing induces on $T^*{\mathfrak g}$
an even supersymmetric nondegenerate bilinear form $B$ which is invariant
($B([X,Y],Z)=B(X,[Y,Z])$ for all $X,Y,Z \in T^*{\mathfrak g}$), i.e. the
structure of a quadratic (or metrised or orthogonal) Lie superalgebra. These
extensions can be classified by the third even scalar cohomology group of
${\mathfrak g}$. Moreover, we show that all finite-dimensional quadratic Lie
superalgebras ${\mathfrak a}={\mathfrak a}_{\bar{0}} \oplus {\mathfrak
a}_{\bar{1}}$ which are either nilpotent, or solvable and such that
$[{\mathfrak a}_{\bar{1}},{\mathfrak a}_{\bar{1}}]\subset [{\mathfrak
a}_{\bar{0}},{\mathfrak a}_{\bar{0}}]$ can be constructed by means of a
$T^*-$extension in the case of an algebraically closed field of characteristic
zero.
