Step 1: Understanding the requirement of mine surveying.
Mine surveying involves the precise measurement of distances, directions, and angles inside underground and surface mines. Accurate measurement of both horizontal and vertical angles is essential for layout, alignment, and safety of mine workings.
Step 2: Identifying the function of theodolite.
A theodolite is a precision surveying instrument specifically designed to measure both horizontal angles and vertical angles accurately. It is widely used in surface as well as underground mine surveying operations.
Step 3: Analysis of the given options.
(A) Dumpy level: Used only for measuring differences in elevation, not angles.
(B) Plane table: Used mainly for graphical surveying and does not provide high accuracy for angle measurement.
(C) Theodolite: Correct — it accurately measures both horizontal and vertical angles, making it ideal for mine surveying.
(D) Sextant: Used mainly in navigation and astronomy, not suitable for mine surveying.
Step 4: Conclusion.
Since the theodolite is capable of accurately measuring both horizontal and vertical angles, it is the most commonly used instrument in mine surveying.