Bankrupt American Airlines and its dogged suitor, US Airways, announced a merger Thursday that would create the world’s largest air carrier and put 86 percent of domestic air travel in the hands of four big airlines. The merger was approved by the boards of both airlines, ending months of negotiation that began with American giving a frosty response to the initial overture from US Airways. Described as the last airline mega-merger in a decade of consolidation, the marriage would see the US Airways brand consigned to the dustbin of aviation history, joining Pan Am, TWA, Eastern Air Lines, Northwest and Continental.The agreement was first reported by other news organizations; a person familiar with the negotiations confirmed the deal to The Washington Post on Wednesday, and it was formally announced early Thursday. The marriage would rescue American from bankruptcy and put the much smaller US Airways into a partnership with the muscle to compete against Delta Air Lines and United Airlines, both of which have grown through recent mergers. Together with Southwest Airlines, the big low-cost carrier, the foursome would dominate passenger travel in the United States. The merger may lead to new competition at the one major airport where US Airways and American have significant overlap Reagan National Airport. Combined, they control almost 68 percent of the flights at National, a situation that could concern regulators.