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I think they do feel sadness. Two days ago I found pigeon in my bedroom mysteriously. It was so weak unable to fly, her eyes looked so sad and was looking for help. I tried to grab it in my hands but it was walking away. I filled water in a cup and few grains in another cup. It started eating daily, I grabbed it in my hands once and started nourishing her head with fingers. I thought I am gonna keep it as my pet but after 2 days it suddenly flew away from my room. Got her energy back, looked healthy. I felt bit sad but okay it is supposed to be free, I'm happy for that.

That is very nice you were able to help her regain her strength and hang out in her company for a few days. Birds are always happier to be free in the sky and trees so its good she flew away in the end I think. Birds in cages often look sad, that is a fact.