The question asks for the instrument commonly used to measure horizontal and vertical angles in mines. The correct answer is the Theodolite. Let's examine why this is the correct choice and why the other options are not suitable for this purpose:
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Theodolite: Theodolites are precision instruments used for measuring angles in the horizontal and vertical planes. They are essential in surveying, especially in mining, for determining the angles at which tunnels must be excavated. Theodolites consist of a telescope mounted on a base, which can be rotated to measure angles with high accuracy. This makes them ideal for tasks requiring precise angle measurement, such as mine planning.
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Dumpy Level: A dumpy level is used primarily for leveling in surveying and construction, measuring horizontal levels rather than angles. It cannot measure vertical or horizontal angles, making it unsuitable for the described purpose in mines.
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Plane Table: A plane table is used for plotting sketches and maps in the field. It is useful for drawing topographical maps but not for accurately measuring angles.
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Sextant: A sextant is traditionally used in navigation to measure the angle between an astronomical object and the horizon. While it measures angles, it is not used in surveying or mine planning to measure horizontal and vertical angles.
Therefore, the theodolite is the most suitable instrument for measuring both horizontal and vertical angles in mine planning, owing to its design and precision. The other options serve different purposes in surveying and navigation.