Methods to prolong a straight line by theodolite:

1. Direct prolongation:
In this method, the theodolite is set up at one end of the line to be prolonged. The line is then sighted through the telescope, and a target is set up at the desired point on the line. The theodolite is then adjusted until the crosshairs of the telescope are aligned with the target. This method is simple and straightforward but may not be suitable for long distances or when obstacles are present.

2. Indirect prolongation:
In this method, the theodolite is set up at a convenient point near the line to be prolonged. A back sight is taken on a point on the line, and the theodolite is then moved to another point where a fore sight is taken on the same line. By repeating this process, the line can be prolonged to the desired distance. This method is more accurate and versatile than direct prolongation, especially for longer distances or when obstacles are present.

3. Double right angles:
In this method, two right angles are established by setting up the theodolite at two points on the line to be prolonged. By bisecting these right angles with the telescope, a perpendicular line can be established, which can then be used to prolong the original line. This method is useful when the line needs to be prolonged at a right angle to its original direction.
Therefore, there are 3 methods commonly used to prolong a straight line by theodolite, each with its own advantages and applications.