This article proposes to conduct natural language inference with novel Enhanced-Relation-Head-Dependent triplets (RHD triplets), which are constructed via enhancing each word in the RHD triplet with its associated local context. Most previous approaches based on deep neural network (DNN) for this task either perform token alignment without considering syntactic dependency among words, or directly use tree-LSTM to generate passage representation with irrelevant information. To improve token alignment and inference judgment with word-pair-dependency, the RHD triplet structure is first proposed. To avoid incorporating irrelevant information, this proposed approach performs comparison directly on each triplet-pair of the given passage-pair (instead of comparing each triplet in a passage with the content merged from the whole opposite passage). Furthermore, to take local context into consideration while conducting token alignment and inference judgment, we also enhance the words of the triplets with their associated local context to improve the performance. Experimental results show that the proposed approach is better than most previous approaches that adopt tree structures, and its performance is comparable to other state-of-the-art approaches (however, our approach is more human comprehensible).