The Pfaffian method enumerating perfect matchings of plane graphs was discovered by Kasteleyn. We use this method to enumerate perfect matchings in a type of graphs with reflective symmetry which is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (1997) 67, MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) If G is a reflective symmetric plane graph without vertices on the symmetry axis, then the number of perfect matchings of G can be expressed by a determinant of order | G|/2, where | G| denotes the number of vertices of G. (2) If G contains no subgraph which is, after the contraction of at most one cycle of odd length, an even subdivision of K 2,3, then the number of perfect matchings of G× K 2 can be expressed by a determinant of order | G|. (3) Let G be a bipartite graph without cycles of length 4 s, s∈{1,2,…}. Then the number of perfect matchings of G× K 2 equals ∏(1+ θ 2) m θ , where the product ranges over all non-negative eigenvalues θ of G and m θ is the multiplicity of eigenvalue θ. Particularly, if T is a tree then the number of perfect matchings of T× K 2 equals ∏(1+ θ 2) m θ , where the product ranges over all non-negative eigenvalues θ of T and m θ is the multiplicity of eigenvalue θ.