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.
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.
Given this definition, then the analogy
teach : teaglach = house : household
would seem to apply — at least as a way of remembering this word.
Prepositions are never able to be translated from one language to another with any consistency