Final answer:
A theodolite is capable of measuring horizontal and vertical angles and is used to check if structures are plumb. The Brunton Compass and sextant are other instruments for measuring angles associated with geological features and celestial bodies, respectively.
Step-by-step explanation:
The instrument that can measure both horizontal angles and vertical angles, and is useful in determining if a post or wall is plumb, is known as a theodolite. This tool is essential in surveying and has a rotating telescope to measure angles in both the horizontal and vertical planes. Surveyors use theodolite to measure azimuth, which are angles measured clockwise from north, with specific azimuth angles corresponding to the cardinal directions (north, east, south, west).
A similar instrument called a Brunton Compass is used by geologists to measure the orientations of geological features with precision. For navigation and astronomical observations, a sextant is utilized to measure angles between celestial bodies and the horizon. Simple tasks like hanging a banner on a wall can be done by using a plumb line or a spirit level to ensure accuracy.
For astronomical measurements, historically, an instrument like a cross-staff would be used to measure angles between two stars, utilizing the concept of a tangent to calculate angles based on the distance between the eye and the cross-piece.