A graph whose edges can be partitioned into 4-cycles in such a way that each vertex is contained in at most two 4-cycles is called a cellular graph. The author gives a ``complementation theorem'' for enumerating the matchings of certain subgraphs of cellular graphs. He applies this theorem to obtain a number of results which include a new proof of Stanley's version of the Aztec diamond theorem, a weighted generalization of a result of Knuth which deals with spanning trees of Aztec diamond graphs, a combinatorial proof of Yang's enumeration of matchings in fortress graphs, and proofs for certain identities of Jockusch and Propp which deal with the numbers of matchings of three kinds of quartered Aztec diamonds.