Lets see, in red picture case, you have 4 triangles to display: upper left half of picture, flipped part of papre that is dragged by finger, and 2 triangles that make remaining quad.
Because you probably want to drag that folder paper at any angle, premade shapes and animations will not work.
You either can create meshes in runtime, or use square meshes with clever and tricky masking.
For squares and masking:
First make algorithm that splits your paper in triangles. Calculate correct uvs for visible part of paper for current triangle.
Then rotate and pan uvs for material. But this all will be hacky way and probably lots of unnecessary problems
So you are really left with making custom meshes. Those are just triangles, biggest problem will be giving them correct uv coordinates.