A TRACON Controller reported their trainee released an aircraft for departure from a non-towered airport utilizing a procedure which placed the aircraft below the Minimum Vectoring Altitude.

Narrative

[I was] training on N/K; south flow; [and] MVFR weather. Aircraft X called for their clearance ZZZ - ZZZ1. The trainee issued the clearance via radar vectors; intending to utilize local Diverse Vector Area procedures as opposed to an as filed" clearance off the ground at ZZZ. In previous discussions with this trainee; I had indicated that as a CPC I do not utilize Diverse Vector Areas because the procedures have changed so much and so frequently and can be different for specific runways; even at the same airport. As such; I had set the expectation that if the trainee intended to utilize Diverse Vector Areas; they would need to show me that they were following the appropriate local rules.The trainee issued an initial heading of 160 degrees to the aircraft and did so with such confidence that I did not want to interrupt the clearance or readback to ensure the heading was legal by our local Diverse Vector Areas procedures. Immediately after the readback the trainee released the aircraft. After this; I queried the trainee to have them show me that they were using the Diverse Vector Areas rules appropriately. At this point; upon looking at the references in the IDS system; we determined that for Runway XX at ZZZ the 160-degree heading was not valid; and should have been at least a 166-degree heading clockwise through a 324-degree heading. Aircraft X subsequently departed and climbed above all applicable MVAs while within 10 NM of ZZZ; as required by the local procedures. The aircraft then left the airspace and continued without further incident.I had too much confidence in my trainee regarding procedures that I am not entirely familiar with myself. I should have either required the trainee to issue the clearance "as filed" or utilizing the published ODP (Obstacle Departure Procedures) for ZZZ. Otherwise; I should have familiarized myself with the appropriate headings for ZZZ for the Diverse Vector Areas before the trainee issued the clearance or released the aircraft. I will exercise additional vigilance in this regard in future training sessions."