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This study explores the flow properties of a couple stress fluid with the consideration of variable viscosity and a uniform transverse magnetic field. Under the effect of irreversible heat transfer, a steady fluid flow has taken place between two parallel inclined plates. The fluid flows due to gravity and the constant pressure gradient force. The plates are fixed and isothermal. The governing equations have been solved analytically for velocity and temperature fields. The total rate of heat flow and volume flow across the channel, skin friction, and Nusselt number at both plates are calculated and represent the impacts of relevant parameters through tables and graphs. The findings show that velocity, temperature, and the total rate of heat flow across the channel are enhanced by increasing the couple stress parameter and the viscosity variation parameter, while increasing the values of the Hartmann number reduces them.