The structures that develop in deformed competent single layers are well known in comparison to the results of the deformation of incompetent single layers. It is argued that the oblique foliations within some sets of deformed igneous sheet intrusions are the most common indication that such sheets deformed as single layers with less competence than their country rocks. Previous workers have implicitly assumed this situation when interpreting examples of such foliated sheets as having sheared between rigid country rocks while still hot. This is taken here as a special case of a much more general situation in which strains in the sheets are induced by strains of the more competent country rocks. As in previous models, the foliation within sheets is interpreted as being oblique to their contacts because of layer-parallel shears along the sheets. In a departure from earlier approaches, such shears are assumed here to arise from the rotation of the sheets due to the bulk strains of their country rocks. This idea is explored using the patterns of the structural movement paths traced by their poles during strain; each type of bulk strain leads to a different pattern of structural movement paths. As the rotation of the sheet contacts drives the rotation of the foliation between them, their poles follow colinear but opposed movement paths and are taken to lie at different positions along what will be called apparent structural movement paths. This working hypothesis is empirically tested using data from two field areas. Sets of apparent structural movement paths for foliated intrusive sheets in two areas are found to compare favorably with theoretical structural movement paths expected if the sheets deformed as incompetent single layers in homogeneously deforming country rocks. The approach developed here can be used to determine empirically the homogeneity of strain of the rocks from which the data have been derived, the orientations and relative values of the principal strains, and an approximation of the shape of the strain ellipsoid. However, no attempt is made to quantify the amount of strain recorded in the field examples quoted here.