bam

I live in Texas and I'm allowed to shoot a gator in September since i live in a ''core county''. Anyways my deer lease aint got no gators on it so I was just wanderin if this would be considered legal. Texas Parks and Wildlife states that a legal method is to use ''hook and line''. LINES SET MUST BE SECURED ON PRIVATE PROPERTY. with that said going further down the rules i read. FIREARMS MAY BE USED TO TAKE ALLIGATORS ONLY ON PRIVATE PROPERTY. IT IS UNLAWFUL TO TAKE AN ALLIGATOR BY MEANS OF FIREARMS FROM, IN, ON, ACROSS OR OVER PUBLIC WATER. Then, An alligator, in public water, caught on a taking device lawfully attached to private property, may be dispatched with a firearm. So my question is, If my property ends at a public lake, would I be allowed to secure my hooking device on my property and catch gators from the lake, bring them onto my property and shoot them?