A new method of obtaining exact, explicit solutions for the diffraction of sound waves by truncated wedges is given which, in conjunction with a previous study [I. Tolstoy, IEEE J. Ocean Eng., in press (1989)], can be applied to problems involving two parallel vertices and arbitrary wedge angles θw, θ′w at these vertices. The method combines exact, explicit representations of the field diffracted by a single vertex with the classic multiple scatter algorithm to formulate the interaction between the line vertices. The previous study had demonstrated the method only for cases in which θw, θw were irrational fractions of π, as well as for the rather special case θw=θ′w =2π, i.e., for the hard strip. In the latter case, the calculations were carried out for wavenumbers k such that 1≲kl≤16, where l was the strip width, and were shown to agree with experimental measurements [H. Medwin et al., J. Acoust. Soc. Am. 72, 1005–1013 (1982)]. The present article extends the theory to cover the important remaining cases for which θ, θw are rational fractions of π; i.e., they are of the form πp/n where p,n are integers. The resulting exact formulas, in combination with previously published ones, thus constitute a complete, exact plane-wave diffraction theory for truncated wedges of any angles θw, θ′w. The formulas have now been applied to the case of a thick rectangular barrier, in the wavenumber range 0.2≲kl≤2.8 and it is shown that the agreement with Medwin et al.’s experimental data is good. The method is quite general and can be extended to more complicated polygonal and broken-line profiles with any number of vertices. As a consequence, there is now a technique for calculating diffraction by realistically shaped seamounts in underwater acoustics as well as by sound barriers in air.