Azimuthal angle (Delta phi) correlations are presented for charged hadrons from dijets for 0.4 < p(T)< 10 GeV/c in Au+Au collisions at root s(NN)=200 GeV. With increasing p(T), the away-side distribution evolves from a broad and relatively flat shape to a concave shape, then to a convex shape. Comparisons to p+p data suggest that the away-side can be divided into a partially suppressed "head" region centered at Delta phi similar to pi and an enhanced "shoulder" region centered at Delta phi similar to pi +/- 1.1. The p(T) spectrum for the head region softens toward central collisions, consistent with the onset of jet quenching. The spectral slope for the shoulder region is independent of centrality and trigger p(T), which offers constraints on energy transport mechanisms and suggests that it contains the medium response to energetic jets.