To interrupt a conversation by saying something suddenly or to insert a remark into what someone else is saying.
From Latin 'interjicere,' where 'inter-' means 'between' and 'jacere' means 'to throw.' So you're literally 'throwing between' words—inserting your comment into the space of someone else's speech.
The word 'interjection' in grammar—those little words like 'oh,' 'wow,' or 'yikes'—comes from this same root, because they're literally thrown between other words to express emotion.
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