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to exchange (messages, greetings, arguments, etc.)
to intersect; to cross; to interlace
... with one another; ... to each other
二人は固い握手を交わして別れを告げた。
The two exchanged a firm handshake and said their goodbyes.
会議の前に意見を交わす時間が設けられた。
Time was set aside to exchange opinions before the meeting.
契約書を交わしてから工事が正式に始まった。
Construction officially began after the contract was signed.