Theodolite traverse ABCDA with missing VCR (68°30′10″)

Assumption: the traverse ABCDA is a closed four-sided figure, so the sum of the interior angles must be 360°.

Method. For each target the mean direction is found from the two faces:

$$\text{Mean direction} = \frac{\text{Face L} + (\text{Face R} \pm 180^\circ)}{2}$$

The angle at a station = mean direction of the forward station - mean direction of the back station. For a closed traverse of $n$ sides, the sum of the interior angles must be $(n-2) \times 180^\circ$; the difference is the angular error, which is distributed equally among the angles (the angles are measured with the same instrument and care).

Field book and mean angles

Check and adjustment

Required sum = $(4-2) \times 180^\circ$ = 360°00′00″. Observed sum = 360°03′20″. Angular error = 360°03′20″ - 360°00′00″ = +0°03′20″. Correction per angle = -0°00′50″ (equal for all 4 angles).

Answer: corrected horizontal angles: A = 70°11′50″, B = 110°24′55″, C = 89°07′55″, D = 90°15′20″.

Missing VCR readings

Missing VCR. For a vertical circle graduated as zenith angle, the sum of the face-left and face-right readings of the same target is 360° when the instrument has no index error. So the missing reading = 360° - the given reading. The vertical angle is $90^\circ - Z$ (+ elevation, - depression).