Today a crazy woman in Union Square tried to make friends with me, the Mouse and Mads. Now, we’re friendly folk. But we have to draw the line somewhere. Here’s where we draw it:
Crazy Lady: Hello!
Mouse, Mads and Me: (Chewing.)
Crazy Lady: I said hello!
Me and Mads: (Looking around.)
Crazy Lady: You don’t got to be antisocial!
Mouse: Um, hi.
Crazy Lady: That’s what I’m sayin’! What are you having for lunch?
Mouse: Chicken.
Crazy Lady: Chicken! That’s good! You know what I’m having? Wendy’s!
Mouse: Oh. Good.
Crazy Lady: Well, I got to go now. You want the rest of my sandwich?
Mouse: No, it’s cool. We’re good.
Crazy Lady: Enjoy your chicken!
I hope her Wendy’s had lithium sprinkled over it.
Jennie Smash 
Non-Interference Pact: Don’t speak to strangers (unless to say something on or near them is on fire).
Right of Privacy: Don’t make eye contact with strangers for more than 2 seconds (unless you’re trying to see what the smoke is coming from).
Right to Construct and Defend One’s Personal Bubble: Except for crowded subways, don’t get within arm’s reach of a stranger (unless putting out the fire).
And most importantly is the principle of Habeas Hamburger: Don’t offer half-eaten food to people who already have some.
I’m sure there were at least a half dozen homeless folk within throwing distance of that crazy sandwich lady. Why didn’t she hurl it at one of them or give it to the winged rats?
Each and every person in public must respect the social contract. An individual’s failure to abide removes any and all obligation required of the offended, such as ‘common’ courtesy or honest directions to the nearest restroom. Extreme violations may be resolved through extreme retaliation. In your specific case, throwing a few pieces of chicken would have been acceptable.