: Is there a technical term for a rhyme using both the end of one sentence, and the beginning of the next? Is there a word for when the last word of a line is the first word of a sentence,

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I'm not sure about the rhyming part, but running a sentence onto the next line is called enjambment:

Enjambment is the continuation of a sentence or clause over a line-break. If a poet allows all the sentences of a poem to end in the same place as regular line-breaks, a kind of deadening can happen in the ear, and in the brain too, as all the thoughts can end up being the same length. Enjambment is one way of creating audible interest; others include caesurae, or having variable line-lengths.

Wiki:

incomplete syntax at the end of a line; the meaning runs over from one poetic line to the next, without terminal punctuation. Lines without enjambment are end-stopped.
In reading, the delay of meaning creates a tension that is released when the word or phrase that completes the syntax is encountered (called the rejet); the tension arises from the "mixed message" produced both by the pause of the line-end, and the suggestion to continue provided by the incomplete meaning. In spite of the apparent contradiction between rhyme, which heightens closure, and enjambment, which delays it, the technique is compatible with rhymed verse. Even in couplets, the closed or heroic couplet was a late development; older is the open couplet, where rhyme and enjambed lines co-exist.

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