We consider the bending degree of freedom for gas phase reactions of the form A + BC → AB + C where A, B, and C are atoms. The potential energy for the bending degree of freedom of the transition state is modeled by the bond-energy-bond-order method and then approximated by a mixed harmonic-quartic potential. The energy levels of the mixed harmonic-quartic potential are calculated by a convenient variation-perturbation method. This treatment is compared to a pure harmonic approximation for the bending potential and shown to be more accurate. The effect of quartic bending anharmonicity on the transition-state rate constants is examined for 72 prototype reactions in the temperature range 200-2400 K. We show that, as compared to the mixed harmonic-quartic treatment, the purely harmonic treatment leads to a significant systematic overestimate of the transition state rate constant. The number of cases, out of the 72 examined, for which the overestimate due to the harmonic approximation exceeds 50% is 8 at 200 K, 23 at 1000 K, and 38 at 2400 K. We also include a detailed study of bending-rotational coupling and a treatment of the H + H2 reaction by using more accurate potential energy surfaces.