A sextant measures vertical angles between two points or between some point and the horizon. It is a combination of telescope, mirrors, and a scale to measure the angle. By measuring the angle of the sun above the horizon at noon, you can determine your latitude on the Earth. William Logan won an octant, a less advanced version of a sextant, as a mathematics prize in university.
A micrometer is used with a telescope to get very accurate measurements of small distances, objects and angles.
A theodolite measures horizontal and vertical angles. It has a telescope that can swing from side to side and up and down. A set of scales measures the angles of movement as the telescope rotates from one position to another. Using trigonometry, a surveyor can sight the same spot from two different locations to get the horizontal and vertical coordinates of that spot. Doing this over a wide area allows a surveyor to map our three-dimensional landscape.