ZLC Controller reported TRACON's non compliance with Center control instruction; resulted in handoff violating Center's MVA and a CFTT event. ZLC Controller also stated that TRACON and ARTCC MIA's are not the same.

Narrative

I was the Radar Controller working Sector 6. Big Sky Approach initiated handoff on Aircraft X and called for coordination. The Big Sky Approach Controller agreed Aircraft X at 120 and deviating right of course. At the time the aircraft was withing 3 minutes of a 126 MIA. I said (something to the effect); 'I need him at thirteen or 10-15 degrees farther right for terrain...radar contact.' About 3 minutes later (now Aircraft X is inside my 126 MIA) the BS Approach Controller called and said; 'he couldn't do thirteen and your control.' I said; 'I need him at thirteen he is in my MIA.' He said; 'I already shipped him; your control.'A minute or so later; I initiated comms with Aircraft X and told him my minimum IFR altitude is 126; climb and maintain one three thousand...and a conversation ensued...including a low altitude alert and an eventual climb from Aircraft X to 130.I believe there are several changeable factors...One would be that Center and the Approach Control have the same MIA's...or at least that they know what ours are before they call when it's too late for us to do anything.The Big Sky Approach Control didn't seem to understand there was a need here even though I gave a control instruction.The pilot of Aircraft X didn't seem to understand there was a need here even though a control instruction was issued.I could have said immediately to the controller to show urgency. I also could have said keep Aircraft X out of my airspace until clear of the MIA...I didn't think that was necessary at the original call.