Mean angles, traverse ABCDA (2076)

Assumption: closed traverse ABCDA with four stations; angle sum must be 360°. Readings are booked in the standard field book form (instrument station, target station, face L, face R, mean direction, angle).

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°02′55″. Angular error = 360°02′55″ - 360°00′00″ = +0°02′55″. Correction per angle = -0°00′43.75″ (equal for all 4 angles).

Answer: corrected horizontal angles: A = 110°24′56.25″, B = 90°15′31.25″, C = 89°07′36.25″, D = 70°11′56.25″.