Started round annoyed (4,3)

I believe the answer is:

teed off

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

This clue could be a double definition.

'started' is the first definition.
The definition and answer can be both related to competing as well as being past participle verbs.
Maybe they are linked in a way I don't understand?

'annoyed' is the second definition.
Although both the answer and definition are past participle verbs, I cannot understand how they can define each other.

'round' acts as a link.

Can you help me to learn more?

I've seen this clue in The Irish Times.

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