How can I cut the end of a line at an angle?

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I have cut a shape and need the ends of the line that create the shape to lean at same angle the shape leans. I can achieve this by overlaying a white image on the line, but I need the entire piece of artwork to have a transparent background, not white. Anyone know how to do this without overlaying an opaque image?

See attached images.

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If your starting shape is a round-cornered parallelogram with a stroke but no fill, you need to use the ‘Expand Stroke’ command to convert it into a filled shape which mimics the stroke. You can then simply subtract the white or grey parallelograms from that shape.

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That did the trick! 

Gracias