Solution Analysis

The problem involves calculating the correction for a specific line in a closed traverse based on a given closing error. A closed traverse ensures that the survey path returns to the starting point. The closing error here is +0.5°, originating from the bearings of the last line, indicating a misclosure in the angle measurements.

• Closing Error: +0.5°
• Number of Sides ($n$): 5
• Line in Question: 4th line ($i=4$)

The total angle misclosure needs to be compensated. The total correction required for the angles is the negative of the closing error.

• Total Correction Required: $-0.5^\circ$

Different methods exist for distributing this correction. Assuming a method where the correction is proportional to the sequence number of the line is required to match the options provided.

Angle Correction Calculation

The correction ($C_i$) for the $i$-th line is calculated proportionally based on its sequence number ($i$) relative to the total number of sides ($n$). The formula used, derived from analyzing the options and the likely intended method, is:

$ C_i = \frac{i}{n} \times (\text{Total Correction Required}) $

For the fourth line ($i=4$) in a five-sided traverse ($n=5$):

$ C_4 = \frac{4}{5} \times (-0.5^\circ) $

$ C_4 = -0.4^\circ $

Convert the correction from decimal degrees to degrees and minutes:

$ -0.4^\circ = -0.4 \times 60 \text{ minutes} = -24 \text{ minutes} $

Thus, the correction for the fourth line is -0°24′.