In this paper, we study cyclotron-resonance linewidths in an electron-phonon system with the presence of a z-directional triangular potential and a y-directional parabolic potential. A general analytical expression for the absorption power is obtained in the presence of an external static magnetic field applied to the sample along the z-axis. The dependence of the absorption power on the photon energy is numerically calculated and graphically plotted for a specific GaAs/AlAs quantum wire. From graphs of the absorption power, we identified the locations of the cyclotron resonance peaks and obtained the cyclotron resonance linewidth (CRLW) as profiles of the curves. The temperature and the magnetic field dependences of the CRLW have been investigated. The CRLW is shown to increase with increasing temperature and magnetic field.