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I understand that がち can be loosely translated as "apt to" "tend to"

and these meanings are present in がち

よく、すぐ、しやすい、することが多い

I can see the meaning of "tend to" easily in this type of sentence

> ラッシュアワーはバスが遅れ **がち** だ。The buses tend to be late during the rush hour.

But when I see the structure がち + に + Verb, I'm not so confident on how these
meanings work. It sounds somewhat strange when I translate it literally.

> 「これを食べてもいいですか」女の子がためらい **がちに** 聞いた。"Can I eat this?" the girl asked with a
> tendency to hesitate. / "Can I eat this?" the girl asked hesitantly.

ためらいがち - tend to hesitate ----> ためらいがちに聞く - ask with a tendency to
hesitate?/ask hesitantly

> 彼は遠慮 **がちに** ドアの後ろに立っていた。He stood behind the door with a tendency to
> hesitate. / he stood behind the door hesitantly.

遠慮がち - tend to hesitate ----> 遠慮がちに立つ - stand with a tendency to
hesitate?/stood hesitantly

So would this sentence I made up be correct?

> 彼は怒り **がちに** ドアを開けた。 He opened the door with a tendency to be angry. / he
> opened the door angrily.

Can someone please clarify and explain if I'm understanding this correctly?
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2019-09-01T11:57:59.760
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question
grammar meaning particles
What is the meaning of がち + に + Verb?
Just weighing in to say that I found [this entry regarding
ためらいがちに](https://www.weblio.jp/content/%E8%BA%8A%E8%BA%87%E3%81%84%E5%8B%9D%E3%81%A1%E3%81%AB):

> 思い切ることができず、半ば思いとどまりながら物事を行うさま。躊躇しつつ行う様子。

"To perform an action with the underlying thought of giving it up; not being
able to make up one's mind completely. To perform an action hesitantly."

So I'd say you're correct in your assessment. Somebody else may answer you on
your made-up sentence though. ;)
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This a very restricted use of `がち`.

### Typical `がち` meaning

Your basic understanding of `がち` as a "接尾語" is correct: it indicates a
tendency or something happening often, and can be constructed with a host of
base words, as in `病気がち`, `遅れがち`, etc.  
It can, grammatically speaking, and in this meaning, be suffixed to any word.
Which ones are acceptable or not is only a question of usage: for example,
`疲れがち` is grammatically correct but in usage `疲れ気味` is usually preferred.

### `がちに` special case

Now, as you mention, `遠慮がち（に）` doesn't really fit with that interpretation. It
doesn't have the meaning of 「遠慮が多い」 or 「遠慮する傾向」 (`遠慮しがち` does instead).  
It has the meaning of "as if" instead (here: "as if restraining oneself"). In
japanese: 「遠慮 **しているような感じ** 」 (taken from the referenced site below).

Contrary to the 1st, typical meaning above though, in this meaning it is
highly idiomatic and can't really be constructed grammatically by suffixing
any noun. I gathered `伏し目がち（に）` and `ためらいがち（に）` falling into this category,
but I doubt there are many more.

In these cases, the `がち` construct seems to have the function of filling a
semantical hole no other established construct can fill: it expresses a link
between two semmingly incompatible actions. In the following examples, at the
face value of the word meanings, it should either be one or the other: 「遠慮する」
or 「言う」, 「伏し目になる」 or 「相手を見る」. Not both together. Hence, when these actions are
associated together using `がちに`, its possible translation of "as if".

> 彼女は遠慮がちに言った
>
> 彼女は伏し目がちに相手を見た

### `怒りがちに` ?

Finally, to answer your last question:

  * 「怒りがちだ」 exists and is correct, but fits the first, typical meaning of `がち` (tendency)
  * 「怒りがちにドアを開けた」 sounds unlikely to me, as the 2 actions are in no way seemingly incompatible ; something like 「怒りがちに無言になった」 maybe, we could consider. However:
  * 「怒りがちに」 is simply not established in usage, so it makes it incorrect, and likely to confuse the person you're speaking to. Furthermore:
  * 「怒りっぽく」 exists and fits the meaning already, and
  * To say "he opened the door angrily" 「怒ってドアを開けた」 is sufficient.

Thinking of the restricted few established uses of `がちに` as having the meaning
of "as if" may further help understanding why it wouldn't be warranted in this
example.

* * *

_Reference:[接尾語「〜がち」について分析](https://nihon5kyoushi.com/2018/08/10/%E6%8E%A5%E5%B0%BE%E8%AA%9E%E3%80%8C%E3%80%9C%E3%81%8C%E3%81%A1%E3%80%8D%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E5%88%86%E6%9E%90/)_
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