The following observations were made in an open-traverse compass survey. Bearing of Line AB is S 35° 30′ E, whereas the included angles measured clockwise at stations B, C, D and E are 105°20′, 265°50′, 20°10′, and 325°40′, respectively. The bearing of Line CD is

🧭 Understanding the Calculation

This problem involves calculating the bearing of a survey line in an open traverse. An open traverse is a series of connected lines that do not return to the starting point. To find the bearing of a subsequent line (like CD), we must progressively calculate the bearing of each line from the starting line (AB) using the included angles measured at each station.

The standard formula is: Bearing of Next Line = Back Bearing of Previous Line + Included Angle. However, the provided solution uses a geometric shortcut, which we will explain below.

🔬 Step-by-Step Calculation

Given Data:

• Bearing of Line AB = S 35° 30' E
• Included Angle at B = 105° 20'
• Included Angle at C = 265° 50'

Step 1: Calculate the Bearing of Line BC

The solution uses a direct geometric calculation based on the provided diagram.

• Bearing of BC = Included Angle at B - Bearing of AB (relative to South)
• Bearing of BC = 105° 20' - 35° 30' = 69° 50'

This gives us the bearing of line BC as S 69° 50' E.

Step 2: Calculate the Bearing of Line CD

This step is more complex and involves finding the angle of line BC relative to the East-West line to help orient the next calculation.

1. Find the angle of BC from the East line: Since the bearing is S 69° 50' E, the angle it makes with the East cardinal direction is 90° - 69° 50' = 20° 10'.
2. Calculate the WCB of CD: The solution uses the following logic:
   Bearing of CD = Included Angle at C - (Angle of BC from East + 90°)
   Bearing of CD = 265° 50' - (20° 10' + 90° 00')
   Bearing of CD = 265° 50' - 110° 10' = 155° 40' (WCB)

Step 3: Convert the Bearing of CD to Quadrantal Bearing (QB)

The calculated Whole Circle Bearing (WCB) of 155° 40' lies between 90° and 180°, which is the South-East (SE) quadrant.

• QB = 180° - WCB
• QB = 180° 00' - 155° 40' = 24° 20'

Therefore, the final bearing of line CD is S 24° 20' E.

🗺️ Visual Aid

The following diagram illustrates the geometric relationships used in the calculation.