teaghlach is a (uncommon, at least in Connemara) word for "family". Most people would say muintir for a broad sense and clann for immediate. You'll also hear gaolta. It's lenited because of the mo before it.
a chéile has a meaning of "together." With le it's just le chéile, lit. with together/ with each other. You can see it was/is lena chéile in parts still with Tá siad ag siúil lena/le chéile (They are walking together; i.e. they are courting).
Let me guess, Munster is the less contemplated in here (it is not a critique), so "teaghlach" is a Monsterism. Now I've seen "céile", is "chéile" its lenition?
I don't know if teaghlach is really even used in Munster. If anything, I'd say Ulster, cause I think Scottish Gaelic still uses it (don't quote me).
Yes, chéile is. It originally comes from the form a chéile. With le, the a fell out over time.
Thank you. (I won't quote). I've read that is used over histrorical topics.
Given this definition, then the analogy
teach : teaglach = house : household
would seem to apply — at least as a way of remembering this word.
Teaghlach was defined in Dinneen’s dictionary as
a household; a hearth, a fireplace; the fireplace in a forge; a fireside; a family.
To me, that suggests that its “family” definition arose as an abstraction of one of its other meanings.
Prepositions are never able to be translated from one language to another with any consistency