The concept of the transmission line, which has proved so fruitful in electronics generally, is adapted for application to planar dielectric waveguiding structures. The possibility of such structures losing signal strength through radiation, and the difficulty of assessing this power loss, is well known in opto-electronics. Here, it is shown that the slab transmission line may be used to obtain exact solutions for open waveguiding structures that have hitherto been unapproachable by such simple means. A new class of waveguide intersections is discovered. The latter have the property that they consist entirely of guided waves, and there is no general radiation. These simple structures may be regarded as outer solutions, where the inner solution depends on the precise structure of the region of intersection. The new solutions predict the possibility of manufacturing new structures which in principle achieve total transmission around a corner at which a waveguide is reflected from a metal or a dielectric mirror. General applications, and applications to optoelectronics using the concept of the effective dielectric constant are discussed. The solutions also provide accurate controls for numerical methods of solution.