  While the problem of determining whether an embedding of a graph $G$ in
$\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying
whether a given embedding of $G$ is {\it rigid} or not is still a hard task
that usually requires ad hoc arguments. In this paper, we show that {\it every}
embedding (not necessarily generic) of a dense enough graph (concretely, a
graph with at least $C_0n^{3/2}\log n$ edges, for some absolute constant
$C_0>0$), which satisfies some very mild general position requirements (no
three vertices of $G$ are embedded to a common line), must have a subframework
of size at least three which is rigid. For the proof we use a connection,
established in Raz [Ra], between the notion of graph rigidity and
configurations of lines in $\mathbb{R}^3$. This connection allows us to use
properties of line configurations established in Guth and Katz [GK2]. In fact,
our proof requires an extended version of Guth and Katz result; the extension
we need is proved by J\'anos Koll\'ar in an Appendix to our paper.
  We do not know whether our assumption on the number of edges being
$\Omega(n^{3/2}\log n)$ is tight, and we provide a construction that shows that
requiring $\Omega(n\log n)$ edges is necessary.
