Argument mining, a subfield of natural language processing and text mining, is a process of extracting argumentative text portions and identifying the role the selected texts play. Legal argument mining targets the argumentative parts of a legal text. In order to better understand how to apply legal argument mining as a step toward improving case summarization, we have assembled a sizeable set of cases and human-expert-prepared summaries annotated in terms of legal argument triples that capture the most important skeletal argument structures in a case. We report the results of applying multiple machine learning techniques to demonstrate and analyze the advantages and disadvantages of different methods to identify sentence components of these legal argument triples.