It is a little bit different in GIMP, and actually easier.
You basically want to rotate the image UNDER the mask.
Here is PS tut
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Here is the GIMP tutorial
Open your image
Create a new black layer
Example in GIMP with a layer mask full white opacity with a triangular shaped black fill for transparency to the bottom layer. Get the triangular shape from your template. Select the mask on the layer and fill with black. This creates your transparent window.
Rotate or move the image however you need.
Now when you got the colors or pattern you want.
Right click the mask layer and select apply layer mask. This places your pie area on a black area.
Select the black color
Select the image layer under the mask layer and right click and select layer to image size. Then delete. You may also have to add an alpha channel before deleting the selected area around the pie shape.
Select>none
Now delete the mask layer.
Select the image layer - layer to image size - and do a Layer>Autocrop layer
You now have a piece of pie to copy and flip however you want.