We have used the Australia Telescope Compact Array (ATCA) to undertake the first high angular resolution observations of 37.7-GHz ($7_{-2} - 8_{-1}E$) methanol masers towards a sample of eleven high-mass star formation regions which host strong 6.7-GHz methanol masers. The 37.7-GHz methanol sites are coincident to within the astrometric uncertainty (0.4 arcseconds) with the 6.7-GHz methanol masers associated with the same star formation region. However, spatial and spectral comparison of the 6.7- and 37.7-GHz maser emission within individual sources shows that the 37.7-GHz masers are less often, or to a lesser degree co-spatial than are the 12.2-GHz and 6.7-GHz masers. We also made sensitive, high angular resolution observations of the 38.3- and 38.5-GHz class II methanol transitions ($6_{2} - 5_{3}A^{-}$ and $6_{2} - 5_{3}A^{+}$, respectively) and the 36.2-GHz ($4_{-1} - 3_{0}E$) class I methanol transition towards the same sample of eleven sources. The 37.7-, 38.3- and 38.5-GHz methanol masers are unresolved in the current observations, which implies a lower limit on the brightness temperature of the strongest masers of more than $10^6$K. We detected the 38.3-GHz methanol transition towards 7 sources, 5 of which are new detections and detected the 38.5-GHz transition towards 6 sources, 4 of which are new detections. We detected 36.2-GHz class I methanol masers towards all eleven sources, 6 of these are new detections for this transition, of which 4 sources do not have previously reported class I methanol masers from any transition.
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