by the way, in this connection, incidentally, in passing
Sentence 1.
Use
明日は図書館が休みです。
The library is closed tomorrow. Incidentally, it is also closed on Mondays.
新しい車を買いました。
I bought a new car. By the way, it was very expensive.
このケーキは手作りです。
This cake is handmade. In this connection, the eggs are from our farm.