To understand the correct answer to the question regarding the micro-optic theodolite and the optical path used for reading angles, let's examine the function and design of a theodolite.

A theodolite is an instrument used in surveying for measuring horizontal and vertical angles. Micro-optic theodolites are a type of theodolite that employ optical systems for precise angle measurements.

• Option 1: Optical and magnified via micrometer eyepiece - This option is correct. In a micro-optic theodolite, the angles are typically read using an optical path that includes a micrometer eyepiece. This system provides magnification and allows for precise measurements of angles. The micrometer eyepiece helps in enhancing the readability of the scale for small increments.
• Option 2: Manual and scale-based - While traditional theodolites might have manual and scale-based readings, micro-optic theodolites generally use optical magnification for improved accuracy.
• Option 3: Satellite via micrometer eyepiece - This option is inaccurate as satellite technology is unrelated to the direct angle measurement mechanisms of a micro-optic theodolite.
• Option 4: Interferometric - Interferometry is a different optical measurement technique often used in high-precision measurements but not typically used in micro-optic theodolites for angle measurement.

Therefore, based on the design and function of micro-optic theodolites, the correct answer is that the optical path used for reading angles is optical and magnified via micrometer eyepiece.