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We saw the deer walking through the woods.
We saw the deer, walking through the woods.

I have learnt if a participle phrase is extraposed, a comma is to precede it to modify the subject: 'we' in my sentence above.
If there is no comma, the participle phrase modifies what comes before (deer).

Let us take a look at the sentence below:

I was irritated by Bill, constantly interrupting.

"In this sentence, interrupting is actually a participle, not a gerund, since it functions as an adjective modifying Bill" Why so?!!!!!!

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Comments  
you don;t need the come in thi case, because the information int he clause sentence (walking through the woods) is not important. That means, that the first part( We saw the deer) can be alone, without the second, it won't hurt the first sentecne construction and grammatic.
InchoateknowledgeWe saw the deer walking through the woods.
We saw the deer, walking through the woods.

I have learnt if a participle phrase is extraposed, a comma is to precede it to modify the subject: 'we' in my sentence above.
If there is no comma, the participle phrase modifies what comes before (deer).

Let us take a look at the sentence below:

I was irritated by Bill, constantly interrupting.

"In this sentence, interrupting is actually a participle, not a gerund, since it functions as an adjective modifying Bill" Why so?!!!!!!

I don't know the answer to your second question, but I have something to say about deer. Emotion: smile

We saw the deer walking through the woods. To me, this means that the deer was (or were) walking in the woods. We might not have been walking, and maybe we weren't even in the woods.

We saw the deer, walking through the woods. But when I see this, I understand it to mean that we walked in the woods and saw the deer. The deer might have been standing still, lying down, running (not necessarily walking).
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Nef, Ksenya, thank you.

I was irritated by Bill, constantly interrupting.

I think this sentence is a compound one with two independent clauses. The second clause is in ellipsis form.

I was irritated by Bill; he was constantly interrupting.

Or

subordinating relation of clauses: because: subordinating conjunction.

I was irritated by Bill, because he was constantly irritating.

It is my vague conjecture.
"In this sentence, interrupting is actually a participle, not a gerund, since it functions as an adjective modifying Bill" Why so?!!!!!!

Participles act as adjectives; gerunds act as nouns.

CJ
InchoateknowledgeI was irritated by Bill, constantly interrupting.

"In this sentence, interrupting is actually a participle, not a gerund, since it functions as an adjective modifying Bill" Why so?!!!!!!

Hi Incho

The end of the sentence is a relative clause equivalent: I was irritated by Bill, who was constantly interrupting.

Cheers
CB
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This is all very well, but then again "irritating" refers to Bill and not me.
Cool Breeze
InchoateknowledgeI was irritated by Bill, constantly interrupting.

"In this sentence, interrupting is actually a participle, not a gerund, since it functions as an adjective modifying Bill" Why so?!!!!!!

Hi Incho

The end of the sentence is a relative clause equivalent: I was irritated by Bill, who was constantly interrupting.

Cheers
CB

very good
Please ignore my last post. I tried to edit it but ran out of time. Sorry.

My short answer:

Compare your sentence with

Interrupting is rude. (Here, interrupting acts as a noun. It isn't modifying anything. It's a gerund.)
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