  A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$
of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb
R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak
quasisymmetry constant of the underlying map for a quasiplane to be a
bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of
bi-Lipschitz images of ${\Bbb R}^n$. One main novelty of these results is that
we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big
pieces criterion, we prove new extension theorems for "almost affine" maps,
which are of independent interest. This work is related to investigations by
Tukia and V\"ais\"al\"a on extensions of quasisymmetric maps with small
distortion.
