Azimuthal angle ($\ensuremath{\Delta}\ensuremath{\phi}$) correlations are presented for a broad range of transverse momentum ($0.4<{p}_{T}<10$ GeV/$c$) and centrality (0--92%) selections for charged hadrons from dijets in Au+Au collisions at $\sqrt{{s}_{\mathit{NN}}}=200$ GeV. With increasing ${p}_{T}$, the away-side $\ensuremath{\Delta}\ensuremath{\phi}$ distribution evolves from a broad and relatively flat shape to a concave shape, then to a convex shape. Comparisons with $p+p$ data suggest that the away-side distribution can be divided into a partially suppressed ``head'' region centered at $\ensuremath{\Delta}\ensuremath{\phi}~\ensuremath{\pi}$, and an enhanced ``shoulder'' region centered at $\ensuremath{\Delta}\ensuremath{\phi}~\ensuremath{\pi}\ifmmode\pm\else\textpm\fi{}1.1$. The ${p}_{T}$ spectrum for the associated hadrons in the head region softens toward central collisions. The spectral slope for the shoulder region is independent of centrality and trigger ${p}_{T}$. The properties of the near-side distributions are also modified relative to those in $p+p$ collisions, reflected by the broadening of the jet shape in $\ensuremath{\Delta}\ensuremath{\phi}$ and $\ensuremath{\Delta}\ensuremath{\eta}$, and an enhancement of the per-trigger yield. However, these modifications seem to be limited to ${p}_{T}\ensuremath{\lesssim}4$ GeV/$c$, above which both the hadron pair shape and per-trigger yield become similar to $p+p$ collisions. These observations suggest that both the away- and near-side distributions contain a jet fragmentation component which dominates for ${p}_{T}\ensuremath{\gtrsim}5$ GeV/$c$ and a medium-induced component which is important for ${p}_{T}\ensuremath{\lesssim}4$ GeV/$c$. We also quantify the role of jets at intermediate and low ${p}_{T}$ through the yield of jet-induced pairs in comparison with binary scaled $p+p$ pair yield. The yield of jet-induced pairs is suppressed at high pair proxy energy (sum of the ${p}_{T}$ magnitudes of the two hadrons) and is enhanced at low pair proxy energy. The former is consistent with jet quenching; the latter is consistent with the enhancement of soft hadron pairs due to transport of lost energy to lower ${p}_{T}$.