Mean direction method, traverse ABCDEA (2080)

Assumption: the title says ABCDEA, but only four instrument stations (A, B, C, D) are given, so the traverse is taken as the closed quadrilateral ABCDA with angle sum 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°02′50″. Angular error = 360°02′50″ - 360°00′00″ = +0°02′50″. Correction per angle = -0°00′42.5″ (equal for all 4 angles).

Answer: corrected horizontal angles: A = 110°24′52.5″, B = 90°15′37.5″, C = 89°07′37.5″, D = 70°11′52.5″.