Cabibbo-allowed charmed meson decays into a pseudoscalar meson and an axial-vector meson are studied. The charm to axial-vector meson transition form factors are evaluated in the Isgur-Scora-Grinstein-Wise quark model. The dipole momentum dependence of the D→ K transition form factor and the presence of a sizable long-distance W-exchange are the two key ingredients for understanding the data of D→K̅a_1. The K_1A-K_1B mixing angle of the strange axial-vector mesons is found to be ≈±37^∘ or ±58^∘

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... from τ→ K_1ν_τ decays. The study of D→ K_1(1270)π, K_1(1400)π decays excludes the positive mixing-angle solutions. It is pointed out that an observation of the decay D^0→ K_1^-(1400)π^+ at the level of 5× 10^-4 will rule out θ≈ -37^∘ and favor the solution θ≈ -58^∘. Though the decays D^0→K̅_1^0π^0 are color suppressed, they are comparable to and even larger than the color-allowed counterparts: K̅_1^0(1270)π^0∼ K_1^-(1270)π^+ and K̅_1^0(1400)π^0> K_1^-(1400)π^+. The finite width effect of the axial-vector resonance is examined. It becomes important for a_1(1260) in particular when its width is near 600 MeV.