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In this paper we give a classification of tilings of the sphere by congruent quadrilaterals with exactly two equal edges. The tilings are the earth map tilings, (p, q)-earth map tilings and their flip modifications, and quadrilateral subdivisions of the cube and the triangular prism. We described the ranges of values of the edges and angles for the tile to be geometrically realisable through extrinsic parameters. The symmetry groups of the tilings are also determined.

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We use rational solutions of trigonometric diophantine equations to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a 3 b). Although the classification is completed in a paper by Cheung, Luk and Yan, the method used here is more systematic and is applicable to other related tiling problems that cannot be solved by the method in the other paper. We also provide detailed geometric data for the tilings.

We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.