Bearing Calculation Steps

The problem asks for the bearing of line BC in the Quadrantal Bearing System (QBS), given the bearing of line AB and the clockwise angle ABC.

Calculate Back Bearing of AB

The given bearing of AB (Whole Circle Bearing - WCB) is $\theta_{AB} = 143^\circ40\prime$. This bearing is measured clockwise from North.

The back bearing of AB (i.e., the bearing of BA) is found by adding $180^\circ$ to the bearing of AB:

$\theta_{BA} = \theta_{AB} + 180^\circ$
$\theta_{BA} = 143^\circ40\prime + 180^\circ$
$\theta_{BA} = 323^\circ40\prime$

Calculate Bearing of BC (WCB)

The angle ABC is given as $309^\circ30\prime$ measured in the clockwise direction. This is the angle measured clockwise from line BA to line BC.

The bearing of BC (WCB) is calculated as:

Bearing of BC (WCB) = Bearing of BA + Angle ABC (clockwise)
Bearing of BC (WCB) = $323^\circ40\prime + 309^\circ30\prime$
Bearing of BC (WCB) = $633^\circ10\prime$

Since a full circle is $360^\circ$, we subtract $360^\circ$ to find the equivalent bearing:

Bearing of BC (WCB) = $633^\circ10\prime - 360^\circ$
Bearing of BC (WCB) = $273^\circ10\prime$

Convert WCB to Quadrantal Bearing System (QBS)

The WCB of BC is $273^\circ10\prime$. This angle falls between $270^\circ$ and $360^\circ$, which corresponds to the North-West (NW) quadrant.

In the QBS, bearings are measured from the North or South meridian towards the East or West.

For the NW quadrant, the bearing is expressed as N (angle) W.

The angle measured from the North meridian towards West is:

Angle = $360^\circ - \text{WCB of BC}$
Angle = $360^\circ - 273^\circ10\prime$
Angle = $86^\circ50\prime$

Therefore, the bearing of line BC in the Quadrantal Bearing System is N $86^\circ50\prime$ W.