At 'hair of the dog,' become angry (7)

I believe the answer is:

bristle

I'm a little stuck... Click here to teach me more about this clue!

'at hair' is the definition.
'bristle' can be an answer for 'hair' (bristle is a kind of hair). I'm not sure about the 'at' bit.

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for bristle that I've seen before include "A short stiff hair, as in beard" , "Get angry" , "Whisker" , "React angrily" , "A stiff hair on chin" .)

I've seen this clue in the Evening Standard.

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