What is the adjusted value of angle D in the least-squares adjustment of the angles?

• A. 129°59'42"
• B. 129°59'38"
• C. 129°59'34"
• D. 129°59'30"

Answer: (B)

In least-squares adjustment of angles, you adjust each measured angle by small corrections so that the overall angular closure constraint is exactly satisfied, while keeping the total squared correction as small as possible. This means the adjusted value for a given angle is the one that best fits the observed readings and the required sum, with adjustments distributed according to the uncertainties of the measurements.

For angle D, the adjusted value that minimizes the overall squared residuals while making the total of the angles meet the known sum ends up at 129°59'38". This choice balances staying close to the measured reading with ensuring the closure condition is exact, using the weights (uncertainties) of the other angles to determine how the corrections are shared. Other possible values would force larger corrections to the angles with tighter precision or violate the closure constraint by a greater amount, increasing the weighted sum of squares.