The present work focuses on the development of a methodology that can be used to solve the inverse kinematics of 6R robot manipulators with Euclidean wrist. Dual quaternions are used as screw motion operators to perform line transformations to the joint axes. The transformations are performed after each joint angle is found. This process accommodates the robot pose on each step from the initial pose to the final pose when all the joint angles have been found. In each step, the appropriate sub problems of Paden-Kahan are used to find the joint angles. The main contribution of the presented methodology is that it is a more general solution than previous approaches. Additionally, an extension of the second sub problem of Paden-Kahan is derived for parallel axes. The proposed algorithm can be used to solve the inverse kinematics problem of most of the 6R robot manipulator configurations, with or without shoulder offset, and with or without intersecting axes of the first and second joint.