In this paper, K quasiconformal maps of Riemann surfaces are investigated. A theorem, which is similar to Schwarz's lemma, is proved for a certain class of K quasiconformal maps. This result is then used to give elementary proofs of theorems concerning K quasiconformal maps. These include Schottky's lemma, Liouville's theorem, and the big Picard theorem. Some of Huber's results on analytic selfmappings of Riemann surfaces are also generalized to the K quasiconformal case. Finally, as an application of the Schwarz type theorem, a geometric proof of a special case of Moser's theorem is given.