The angular misclosure of the closed-loop traverse shown in the figure is ______ ° (Answer in decimal degrees and rounded off to three decimal places).
Show Hint
Sum the six interior angles of the closed traverse and compare with the theoretical sum (n-2) x 180 degrees for an n-sided polygon.
Step 1: Identify the polygon and the angle closure condition. The closed traverse shown has six stations, so the figure is a hexagon (n = 6 sides, with one reentrant/concave vertex where the interior angle exceeds \(180^\circ\)). For any closed polygon traverse, the theoretical sum of the interior angles is \[ \Sigma \theta_{theoretical} = (n-2) \times 180^\circ \] For \(n = 6\): \[ \Sigma \theta_{theoretical} = (6-2) \times 180^\circ = 720^\circ \] Step 2: Add the six observed interior angles. The observed angles read from the figure are \(132^\circ45'30''\), \(64^\circ00'00''\), \(227^\circ26'15''\) (the reflex angle at the concave station), \(97^\circ35'45''\), \(131^\circ35'00''\) and \(66^\circ40'30''\). Adding the seconds: \(30''+00''+15''+45''+00''+30'' = 120'' = 2'00''\). Adding the minutes with the carried 2': \(45'+00'+26'+35'+35'+40'+2' = 183' = 3^\circ03'\). Adding the degrees with the carried \(3^\circ\): \(132^\circ+64^\circ+227^\circ+97^\circ+131^\circ+66^\circ+3^\circ = 720^\circ\), giving \[ \Sigma \theta_{observed} = 720^\circ03'00'' \] Step 3: Compute the angular misclosure. The angular misclosure is the difference between the observed sum and the theoretical sum: \[ e = \Sigma \theta_{observed} - \Sigma \theta_{theoretical} = 720^\circ03'00'' - 720^\circ00'00'' = 0^\circ03'00'' \] Converting \(3'\) to decimal degrees: \(3' = 3/60 = 0.05^\circ\). Step 4: State the result. So the angular misclosure of the traverse is \[ \boxed{e = 0.050^\circ} \]
