The majority of existing work on agent dialogues considers negotiation, persuasion or deliberation dialogues. We focus on inquiry dialogues that allow two agents to share knowledge in order to construct an argument for a specific claim. Inquiry dialogues are particularly useful in cooperative domains such as healthcare, and can be embedded within other dialogue types. Existing inquiry dialogue systems only model dialogues, meaning they provide a protocol which dictates what the possible legal next moves are but not which of these moves to make. Our system not only includes a general dialogue-game style inquiry protocol but also a strategy, for an agent to use with this protocol, that selects exactly one of the legal moves to make. We propose a benchmark against which we compare our dialogues, being the arguments that can be constructed from the union of the agents' beliefs, and use this to define soundness and completeness properties for inquiry dialogues. We show that these properties hold for all well-formed inquiry dialogues in our system.