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The endgame is a delicate part of a chess game where every pawn matters. The key to all checkmates is to keep the opponents king confined. Do not be overanxious to check the king, as it will not work.  First move the rook to the rank ahead of the enemy king. This will confine the king to a certain number of squares. Advance the king to obtain opposition when your king is in front of the opponent's. When he moves away, you will need to make a waiting move, just move the rook one square over. He will move the king away from your king. When the kings are opposite each other, check him with the rook, and then repeat the process until he is on the back rank, where check becomes checkmate. The Queen checkmate is identical, but you must be careful not to stalemate.
Don't discount any pieces during the endgame. Use your rook against the king.