Real mad, but terribly frightened (7)

I believe the answer is:

alarmed

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

'frightened' is the definition.
(alarming is a kind of frightening)

'real mad but terribly' is the wordplay.
'real mad' anagrams to 'ALARMED'.
However, I am not clear how this is indicated.

Can you help me to learn more?

(Other definitions for alarmed that I've seen before include "Fearful" , "Apprehensive" , "In a panic" , "Experiencing a sudden sense of danger" , "Frightened, frightened by danger" .)

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