In a least-squares adjustment of three measured horizontal angles, what is the adjusted value of angle A?

• A. 69°59'14"
• B. 69°59'22"
• C. 69°59'10"
• D. 69°59'18"

Answer: (A)

In least-squares adjustment of horizontal angles, the measured angles are corrected so that they satisfy a known geometric constraint, typically that the three angles around a point sum to a full turn (360 degrees). The goal is to find adjusted values that stay as close as possible to the observed ones while exactly meeting this sum, with corrections distributed according to how reliable (weighted) each angle is.

So the adjusted value for angle A is the result of applying that constraint and the observed data together. If the three measured angles don’t add up to 360 degrees, the adjustment shifts each angle slightly—the amount for A is determined by the balance of the residuals and the weights of the measurements. In this case, the adjustment yields 69°59'14" for angle A because that value, combined with the adjusted B and C, satisfies the 360° sum and minimizes the weighted square of all corrections.

If you compared other possible values, they would either fail to produce the exact closure sum with the other adjusted angles or would produce a larger overall adjustment when considering the measurement reliabilities. The chosen value is the one that best satisfies both the closure condition and the quality of the observations.