Here's a pretty easy and crude way to redneck engineer it, trussing it on the fly if you will.
Replace your 2x2 ridge pole with a 2x4 ridge pole. Just run it along the length standing up on end.
Lay a 2x2 on the top of the sides roughly where you're going to put the roof trusses. This will keep the sides from bowing out under the weight of the roof and especially the roof and snow.
Now we'll make the "truss" on the fly. Just lay a 2x4 up onto the ridgepole and the side where you put the 2x2. Right about the location you drew in red on your last drawing.
You can stand the 2x4 on edge, and make fancy bevel cuts to install it most nicely. Or, you can just whack it into place, laying it flat. Since this is just a small coop, crude will work just fine. But you'll be very proud of yourself if you do the fancier bevel cuts. It's pretty easy to hold the board up and just eyeball a good line.
If you're wondering why I said to place only a 2x2 across the sides, it's because those pieces won't see much load, and it'll be in tension. Now if you're planning to hang things from those oh so convenient cross bars, use a 2x4 instead.
