1. A system for indirectly measuring a geometric dimension related to an opening in an apertured exterior surface of a part based on direct measurements of the part when fixtured at a measurement station, the system comprising: first and second holding devices for holding a part therebetween in a part-retaining position in which the part is firmly held between the devices at the apertured exterior surface wherein a portion of one of the holding devices extends into the opening in the part-retaining position; a head apparatus including a plurality of radiation sources for successively directing a first array of planes of radiation at the one of the holding devices and a second array of planes of radiation at the part in the part-retaining position so that the one of the holding devices and the part occlude their respective planes of radiation to create corresponding arrays of unobstructed planar portions of the planes of radiation wherein each of the unobstructed planar portions contains an amount of radiation which is representative of a respective external geometric dimension of either the one of the holding devices or the part, the head apparatus further including a plurality of receiver modules for measuring the amount of radiation present in each of the unobstructed planar portions to obtain holding device and part signals; a movable stage subsystem coupled to the head apparatus for translating the head apparatus relative to the devices and the part so that the planes of radiation scan the one of the holding devices and the part in the part-retaining position; a signal processor for processing the holding device and part signals to obtain data; and a data processor for processing the data to obtain the direct measurements, the data processor determining the geometric dimension related to the opening based on the direct measurements.