The dog must have been yelping pretty badly I presume, and their may have been severe exterior damage to warrant a gunshot to the head.

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As in humans accidents its not the noisiest patients that usually need urgent treatment first but rather the quite ones.
If the sitrep is true I`m sure the officer used his best judgement call when it was decided to end the animals suffering,this said I still think the officer needs more practice on the firing range.