Question

Based on the theodolite survey for a closed traverse PQRS, the following bearings are observed for the sides of the traverse.

The interior angles at P and R respectively are:

• A. \( 45^\circ 00', 115^\circ 00' \)
• B. \( 70^\circ 00', 65^\circ 00' \)
• C. \( 135^\circ 00', 90^\circ 00' \)
• D. \( 250^\circ 00', 115^\circ 00' \)

Hint:

When calculating interior angles in a closed traverse, use the formula \( 180^\circ + ({Next Bearing} - {Previous Bearing}) \) to find the angle between two consecutive sides.

Answer 1

The Correct Option is B

Step 1: Understand the traverse bearings.
We are given the fore bearings of the lines of the closed traverse PQRS:
\( PQ = 60^\circ 30' \)
\( QR = 105^\circ 30' \)
\( RS = 220^\circ 30' \)
\( SP = 310^\circ 30' \)
To calculate the interior angles at P and R, we need to use the formula for the interior angle at any point in a closed traverse: \[ {Interior Angle} = 180^\circ + ({Next Bearing} - {Previous Bearing}) \] Step 2: Calculate the interior angle at P.
The interior angle at P is formed between lines PQ and SP. The formula for the interior angle at P is: \[ {Angle at P} = 180^\circ + (SP - PQ) \] Substituting the values: \[ {Angle at P} = 180^\circ + (310^\circ 30' - 60^\circ 30') = 180^\circ + 250^\circ = 70^\circ 00' \] Step 3: Calculate the interior angle at R.
The interior angle at R is formed between lines QR and RS. The formula for the interior angle at R is: \[ {Angle at R} = 180^\circ + (RS - QR) \] Substituting the values: \[ {Angle at R} = 180^\circ + (220^\circ 30' - 105^\circ 30') = 180^\circ + 115^\circ = 65^\circ 00' \] Conclusion: The interior angles at P and R are \( 70^\circ 00' \) and \( 65^\circ 00' \) respectively. Answer: The correct option is \( \boxed{(B)} \).