Someone who is engaged in a particular activity is doing that thing.
the various projects he was engaged in.
他参与过的不同项目。
When two people are engaged, they have agreed to marry each other.
We got engaged on my eighteenth birthday.
我们在我18岁生日那天订婚了。
If a telephone or a telephone line is engaged, it is already being used by someone else so that you are unable to speak to the person you are phoning.
If a public toilet is engaged, it is already being used by someone else.