Question

Interior angles measured at the locations of a closed traverse ABCDA are given in table below.

Location	Interior angle
A	71° 1' 40"
B	104° 54' 23"
C	107° 54' 10"
D	76° 20' 42"

The total error in the measured angles (in degrees) is (rounded off to three decimal places).

Hint:

Theoretical angle sum for a closed traverse of \(n\) sides is \((n-2)\times180^\circ\); compare with the measured sum to get the error.

Answer 1

Correct Answer: 0.182

Step 1: Recall the theoretical angle sum for a closed traverse.
For any closed traverse with \(n\) sides, the sum of the interior angles should theoretically equal
\[ \Sigma\theta = (n-2) \times 180^\circ \]
Traverse ABCDA has 4 stations (A, B, C, D) and 4 sides, so \(n=4\):
\[ \Sigma\theta_{theory} = (4-2)\times 180^\circ = 360^\circ \]

Step 2: Add the measured angles.
Sum the degrees, minutes and seconds separately:
Degrees: \(71+104+107+76 = 358\)
Minutes: \(1+54+54+20 = 129\)
Seconds: \(40+23+10+42 = 115\)

Step 3: Convert the raw totals into proper degrees-minutes-seconds.
\(115'' = 1'55''\) (since \(115-60=55\), carry 1 minute). Minutes become \(129+1=130'\), and \(130' = 2^\circ10'\) (since \(130-120=10\), carry 2 degrees). Degrees become \(358+2=360\). So the measured sum is
\[ \Sigma\theta_{measured} = 360^\circ 10' 55'' \]

Step 4: Find the total angular error.
\[ \text{Error} = \Sigma\theta_{measured} - \Sigma\theta_{theory} = 360^\circ10'55'' - 360^\circ0'0'' = 0^\circ10'55'' \]

Step 5: Convert the error to decimal degrees.
\[ 10' = \frac{10}{60} = 0.16667^\circ, \qquad 55'' = \frac{55}{3600} = 0.01528^\circ \]
\[ \text{Error} = 0.16667^\circ + 0.01528^\circ = 0.18194^\circ \]

Final Answer:
Rounded to three decimal places,
\[ \boxed{\text{Error} \approx 0.182^\circ} \]