A technique is presented by way of example for proving the existence of minimal surfaces bounded by straight line segments and planar curves along which the surface meets the plane of the curve at a constant angle. Set in complex analysis, this technique provides a way to construct new examples of soap films and capillary surfaces. The soap films established are a soap film spanning five edges of a regular tetrahedron and a soap film spanning a rectangular prism. The examples of capillary graphs over a square presented here were previously shown to exist by Concus, Finn, and McCuan. However, with this new approach, we are able to examine the behavior of the graphs at the corners of the square. More precisely, we construct two one-parameter families of capillary graphs. The first family provides examples of capillary graphs that have continuous unit normal up to the corner, but the graphing function is not C2 at the corner. The second family consists of capillary graphs with contact angle data in D+2∪D-2 such that the graphing function has a finite jump discontinuity at each corner.