where your 2 angles meet, on the iside of your angle, run your 2 studs to form a vertical -V. then on your front (which is your open part of the V) measure from your stud to the 22.5 angle. take that measurement to another stud and rip it on a 22.5 degee. do that on 2 studs and fit them into the opening to give you a solid corner. thats the easiest way i do it

You don't frame at 22.5 you frame at 45, the top plates get 45's opposing so they over lap.

You have to rip for the corners & pre-insulate.

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For your verticle stud, you actually can do this easier by cutting off the oposite corners lengthwise of a 2x6 and not rip a 2x4. This is sturdier and gives you more to nail into. I mocked up ripping a 2x4 in half and flipping the one stufd and there is nothing substantial to nail to.

-Dave