Strong and virtually leak-free rectangular duct connection systems. Ductmate 35 connection systems consist of roll-formed flanges, corner pieces, gasket, and cleat. The flanges attach to the duct wall and have an integral mastic which allows the flange to seal itself to the duct. Corner pieces are used to add rigidity to the flange, hold the ductwork together, and provide a sealing surface for the gasket. The gasket serves as a seal between the flanges. The cleat insures even compression of the gasket along the length of the flange.