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An Interference Channel with Generalized Feedback (IFC-GF) models a wireless network where the sources can sense the channel activity. The signal overheard from the channel provides information about the activity of the other sources and thus furnishes the basis for cooperation. This two-part paper studies achievable strategies (Part I) and outer bounds (Part II) for the general discrete memoryless IFC-GF with two source-destination pairs. In Part I, the generalized feedback is used to gain knowledge about the message sent by the other source and then exploited in two ways: a) to relay the messages that can be decoded at both destinations, thus realizing the gains of beam-forming of a distributed multiantenna system, and b) to hide the messages that can not be decoded at the nonintended destination, thus leveraging the interference “precancellation” property of dirty-paper coding. We show that our achievable region generalizes several known achievable regions for the IFC-GF and that it reduces to known achievable regions for the channels subsumed by the IFC-GF model. For the Gaussian channel, it is shown that source cooperation enlarges the achievable rate region of the corresponding IFC without generalized feedback/cooperation.