In this paper, we propose to leverage the unique characteristics of dialogues sharing commonsense knowledge across participants, to resolve the difficulties in summarizing them. We present SICK, a framework that uses commonsense inferences as additional context. Compared to previous work that solely relies on the input dialogue, SICK uses an external knowledge model to generate a rich set of commonsense inferences and selects the most probable one with a similarity-based selection method. Built upon SICK, SICK++ utilizes commonsense as supervision, where the task of generating commonsense inferences is added upon summarizing the dialogue in a multi-task learning setting. Experimental results show that with injected commonsense knowledge, our framework generates more informative and consistent summaries than existing methods.