I basically want to change the angle of the red part, but without changing the cut-off location/intersection point with the white part as you see here. When I tried to find a solution to the problem, I would change the angle but that would also change the way that it cuts off the top part of the white part. As you see here, the white wall is perfectly cut. I need to accomplish this.
You have to rotate it around the intersection. To do that you shift the rotated parts cframe so that it has the same position as the intersection, then do your rotation, then shift it back. The shift that you do will be a translation in the space of the rotated part.
It depends which CFrame of the pivot point you already have. Do you have it relative to the rotated part (like an Attachment.CFrame) or relative to the world (like an Attachment.WorldCFrame)?