In the sentence "the door slammed shut with a loud bang," there are elements of both onomatopoeia and alliteration.
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Onomatopoeia: The word "bang" is an example of onomatopoeia because it imitates the sound that is being described.
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Alliteration: The sentence does not contain alliteration because there are no repeating initial consonant sounds in close succession. However, if you were to modify the sentence slightly, you might be able to create alliteration.
So, the sentence contains onomatopoeia but not alliteration.