Dihadron azimuthal correlations containing a high transverse momentum (pT) trigger particle are sensitive to the properties of the nuclear medium created at RHIC through the strong interactions occurring between the traversing parton and the medium, i.e. jet-quenching. Previous measurements revealed a strong modification to dihadron azimuthal correlations in Au+Au collisions with respect to p+p and d+Au collisions. The modification increases with the collision centrality, suggesting a path-length or energy density dependence to the jet-quenching effect. This paper reports STAR measurements of dihadron azimuthal correlations in mid-central (20%-60%) Au+Au collisions at GeV as a function of the trigger particle's azimuthal angle relative to the event plane, . The azimuthal correlation is studied as a function of both the trigger and associated particle . The subtractions of the combinatorial background and anisotropic flow, assuming Zero Yield At Minimum (ZYAM), are described. The correlation results are first discussed with subtraction of the even harmonic (elliptic and quadrangular) flow backgrounds. The away-side correlation is strongly modified, and the modification varies with , with a double-peak structure for out-of-plane trigger particles. The near-side ridge (long range pseudo-rapidity correlation) appears to drop with increasing while the jet-like component remains approximately constant. The correlation functions are further studied with the subtraction of odd harmonic triangular flow background arising from fluctuations. It is found that the triangular flow, while responsible for the majority of the amplitudes, is not sufficient to explain the -dependence of the ridge or the away-side double-peak structure. The dropping ridge with could be attributed to a -dependent elliptic anisotropy; however, the physics mechanism of the ridge remains an open question. Even with a -dependent elliptic flow, the away-side correlation structure is robust. These results, with extensive systematic studies of the dihadron correlations as a function of , trigger and associated particle , and the pseudo-rapidity range , should provide stringent inputs to help understand the underlying physics mechanisms of jet-medium interactions in high energy nuclear collisions.