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When cutting a few millimetres from a piece of glass, instead of scoring the line and nibbling with your grozing pliers, an alternative method is illustrated below: -
After scoring, place a solid wooden straight-edge on the overhanging sliver.
Apply a sharp blow with the lower hand while applying firm pressure with your upper hand.
With practise you should be able to remove a 2mm unbroken sliver from a full sheet of glass using this method.