Besides square, the edges need to be prepped for bonding. I'd be curious to know what the current edge treatment is. If they're flame polished it will probably craze really badly if you try to solvent weld it. Especially if it's low quality extruded junk (which it probably is, if it was shelves at one point).
I'd take it to a plastics shop and ask them to cut the pieces for the ends and run ALL the pieces through a router table to ensure the edges are flat/parallel/ready for solvent welding. This will not be their favorite job of the day (working on someone else's mystery material) so bringing donuts or beer will definitely help.
If you've never worked with acrylic, I'd probably also ask them if they have some random scraps laying around you could practice your bonding on. And get some weldon 3 or 4 and some syringes. And google the pins method for bonding acrylic.