Edit: I don't know if this constitutes "arguing" but a lot of people are suggesting that I was somehow partially at fault because I left the gate open. This boggles my mind because I could have been in exactly the same situation in my front yard, where there is no fence at all. I struggle to see why the state of my gate is somehow the relevant factor for a lot of people.

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YOU left a gate open that YOU STATE THE DOG ENTERED YOUR YARD THROUGH. You stated the issue in your own edit and STILL don't understand why the dog was there outside of the fact that there HAD to be some other way for you to intervene in this situation besides kicking the dog into the air. What is wrong with you?

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YTA.