  We present the detection of [OIII] emission lines from the galaxies
responsible for two high-redshift z>1.75 damped Lyman-alpha (DLA) absorption
lines. These are the first detections of rest-frame optical emission lines from
high-redshift DLA galaxies. Unlike the Lyman-alpha line, the [OIII] line
provides a measure of the systemic velocity of the galaxy. We compare the
[OIII] redshifts with the velocity profile of the low-ionisation metal lines in
these two absorbers, with the goal of distinguishing between the model of
Prochaska and Wolfe of DLA absorbers as large rapidly rotating cold thick
discs, and the standard hierarchical CDM model of structure formation, in which
DLAs arise in protogalactic fragments. We find some discrepancies with the
predictions of the former model. Furthermore the image of the DLA galaxy
towards Q2206-1958 shows a complex disturbed morphology, which is more in
accord with the hierarchical picture. We use the properties of the rest-frame
optical emission lines to further explore the question posed by Moller et al.:
are high-redshift DLA galaxies Lyman-break galaxies (LBGs) selected by gas
cross section? The measured velocity dispersions of the DLA galaxies are in
agreement with this picture, while the data on the [OIII] luminosities and the
velocity differences between the Lyman-alpha and [OIII] lines are inconclusive,
as there are insufficient LBG measurements overlapping in luminosity. Finally
we estimate the star formation rates in these two DLA galaxies, using a variety
of diagnostics, and include a discussion of the extent to which the [OIII] line
is useful for this purpose. (abridged)
