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The calculated bearings for all traverse lines of the polygon are summarized in the table below. The calculations proceed sequentially, starting with the given bearing of line AB and using the measured interior angle at each station. Summary of Traverse Bearings | Traverse Line | Bearing (Forward Bearing) | |---|---| | AB | \mathbf{N \ 41^\circ 35' \ E} (Given) | | BC | \mathbf{N \ 9^\circ 14' \ W} | | CD | \mathbf{S \ 79^\circ 21' \ W} | | DE | \mathbf{S \ 31^\circ 51' \ W} | | EF | \mathbf{S \ 12^\circ 27' \ E} | | FA | \mathbf{S \ 73^\circ 35' \ E} | Detailed Calculation Steps The bearing of the next line is found by: \text{Bearing}_{\text{Next Line}} = \text{Back Bearing}_{\text{Previous Line}} \pm \text{Interior Angle} We use Whole Circle Bearings (WCB) for easy calculation, where \text{WCB} = \text{Quadrant Bearing (QB)} for N/E lines, \text{WCB} = 180^\circ - \text{QB} for S/E lines, etc. The Back Bearing (BB) is always \text{FB} \pm 180^\circ. Since the angles are interior and measured to the right (clockwise) from the back line, we use: \text{WCB}_{\text{Next}} = \text{WCB}_{\text{Back Line}} + \text{Interior Angle} 1. Line AB \to BC * Given \text{FB}_{AB}: \mathbf{N \ 41^\circ 35' \ E} (\text{WCB}_{AB} = 41^\circ 35') * Back Bearing \text{BB}_{AB} at B (\text{WCB}_{BA}): 41^\circ 35' + 180^\circ 00' = 221^\circ 35' * Angle \angle B: 129^\circ 11' * \text{WCB}_{BC}: 221^\circ 35' + 129^\circ 11' = 350^\circ 46' * \text{FB}_{BC} (QB): 360^\circ 00' - 350^\circ 46' = 9^\circ 14' \implies \mathbf{N \ 9^\circ 14' \ W} 2. Line BC \to CD * \text{FB}_{BC}: \mathbf{N \ 9^\circ 14' \ W} * Back Bearing \text{BB}_{BC} at C (\text{WCB}_{CB}): 360^\circ 00' - 9^\circ 14' = 350^\circ 46', then 350^\circ 46' - 180^\circ 00' = 170^\circ 46' * Angle \angle C: 88^\circ 35' * \text{WCB}_{CD}: 170^\circ 46' + 88^\circ 35' = 259^\circ 21' * \text{FB}_{CD} (QB): 259^\circ 21' - 180^\circ 00' = 79^\circ 21' \implies \mathbf{S \ 79^\circ 21' \ W} 3. Line CD \to DE * \text{FB}_{CD}: \mathbf{S \ 79^\circ 21' \ W} * Back Bearing \text{BB}_{CD} at D (\text{WCB}_{DC}): 259^\circ 21' - 180^\circ 00' = 79^\circ 21' * Angle \angle D: 132^\circ 30' * \text{WCB}_{DE}: 79^\circ 21' + 132^\circ 30' = 211^\circ 51' * \text{FB}_{DE} (QB): 211^\circ 51' - 180^\circ 00' = 31^\circ 51' \implies \mathbf{S \ 31^\circ 51' \ W} 4. Line DE \to EF * \text{FB}_{DE}: \mathbf{S \ 31^\circ 51' \ W} * Back Bearing \text{BB}_{DE} at E (\text{WCB}_{ED}): 211^\circ 51' - 180^\circ 00' = 31^\circ 51' * Angle \angle E: 135^\circ 42' * \text{WCB}_{EF}: 31^\circ 51' + 135^\circ 42' = 167^\circ 33' * \text{FB}_{EF} (QB): 180^\circ 00' - 167^\circ 33' = 12^\circ 27' \implies \mathbf{S \ 12^\circ 27' \ E} 5. Line EF \to FA * \text{FB}_{EF}: \mathbf{S \ 12^\circ 27' \ E} * Back Bearing \text{BB}_{EF} at F (\text{WCB}_{FE}): 167^\circ 33' + 180^\circ 00' = 347^\circ 33' * Angle \angle F: 118^\circ 52' * \text{WCB}_{FA}: 347^\circ 33' + 118^\circ 52' = 466^\circ 25' * 466^\circ 25' - 360^\circ 00' = 106^\circ 25' * \text{FB}_{FA} (QB): 180^\circ 00' - 106^\circ 25' = 73^\circ 35' \implies \mathbf{S \ 73^\circ 35' \ E} 6. Line FA \to AB (Closure Check) * \text{FB}_{FA}: \mathbf{S \ 73^\circ 35' \ E} * Back Bearing \text{BB}_{FA} at A (\text{WCB}_{AF}): 106^\circ 25' + 180^\circ 00' = 286^\circ 25' * Angle \angle A: 115^\circ 10' * Calculated \text{WCB}_{AB}: 286^\circ 25' + 115^\circ 10' = 401^\circ 35' * 401^\circ 35' - 360^\circ 00' = 41^\circ 35' * Calculated \text{FB}_{AB} (QB): 41^\circ 35' \implies \mathbf{N \ 41^\circ 35' \ E} * The calculated bearing matches the given bearing, confirming the accuracy of the calculations.

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The calculated bearings for all traverse lines of the polygon are summarized in the table below. The calculations proceed sequentially, starting with the given bearing of line AB and using the measured interior angle at each station. Summary of Traverse Bearings | Traverse Line | Bearing (Forward Bearing) | |---|---| | AB | \mathbf{N \ 41^\circ 35' \ E} (Given) | | BC | \mathbf{N \ 9^\circ 14' \ W} | | CD | \mathbf{S \ 79^\circ 21' \ W} | | DE | \mathbf{S \ 31^\circ 51' \ W} | | EF | \mathbf{S \ 12^\circ 27' \ E} | | FA | \mathbf{S \ 73^\circ 35' \ E} | Detailed Calculation Steps The bearing of the next line is found by: \text{Bearing}_{\text{Next Line}} = \text{Back Bearing}_{\text{Previous Line}} \pm \text{Interior Angle} We use Whole Circle Bearings (WCB) for easy calculation, where \text{WCB} = \text{Quadrant Bearing (QB)} for N/E lines, \text{WCB} = 180^\circ - \text{QB} for S/E lines, etc. The Back Bearing (BB) is always \text{FB} \pm 180^\circ. Since the angles are interior and measured to the right (clockwise) from the back line, we use: \text{WCB}_{\text{Next}} = \text{WCB}_{\text{Back Line}} + \text{Interior Angle} 1. Line AB \to BC * Given \text{FB}_{AB}: \mathbf{N \ 41^\circ 35' \ E} (\text{WCB}_{AB} = 41^\circ 35') * Back Bearing \text{BB}_{AB} at B (\text{WCB}_{BA}): 41^\circ 35' + 180^\circ 00' = 221^\circ 35' * Angle \angle B: 129^\circ 11' * \text{WCB}_{BC}: 221^\circ 35' + 129^\circ 11' = 350^\circ 46' * \text{FB}_{BC} (QB): 360^\circ 00' - 350^\circ 46' = 9^\circ 14' \implies \mathbf{N \ 9^\circ 14' \ W} 2. Line BC \to CD * \text{FB}_{BC}: \mathbf{N \ 9^\circ 14' \ W} * Back Bearing \text{BB}_{BC} at C (\text{WCB}_{CB}): 360^\circ 00' - 9^\circ 14' = 350^\circ 46', then 350^\circ 46' - 180^\circ 00' = 170^\circ 46' * Angle \angle C: 88^\circ 35' * \text{WCB}_{CD}: 170^\circ 46' + 88^\circ 35' = 259^\circ 21' * \text{FB}_{CD} (QB): 259^\circ 21' - 180^\circ 00' = 79^\circ 21' \implies \mathbf{S \ 79^\circ 21' \ W} 3. Line CD \to DE * \text{FB}_{CD}: \mathbf{S \ 79^\circ 21' \ W} * Back Bearing \text{BB}_{CD} at D (\text{WCB}_{DC}): 259^\circ 21' - 180^\circ 00' = 79^\circ 21' * Angle \angle D: 132^\circ 30' * \text{WCB}_{DE}: 79^\circ 21' + 132^\circ 30' = 211^\circ 51' * \text{FB}_{DE} (QB): 211^\circ 51' - 180^\circ 00' = 31^\circ 51' \implies \mathbf{S \ 31^\circ 51' \ W} 4. Line DE \to EF * \text{FB}_{DE}: \mathbf{S \ 31^\circ 51' \ W} * Back Bearing \text{BB}_{DE} at E (\text{WCB}_{ED}): 211^\circ 51' - 180^\circ 00' = 31^\circ 51' * Angle \angle E: 135^\circ 42' * \text{WCB}_{EF}: 31^\circ 51' + 135^\circ 42' = 167^\circ 33' * \text{FB}_{EF} (QB): 180^\circ 00' - 167^\circ 33' = 12^\circ 27' \implies \mathbf{S \ 12^\circ 27' \ E} 5. Line EF \to FA * \text{FB}_{EF}: \mathbf{S \ 12^\circ 27' \ E} * Back Bearing \text{BB}_{EF} at F (\text{WCB}_{FE}): 167^\circ 33' + 180^\circ 00' = 347^\circ 33' * Angle \angle F: 118^\circ 52' * \text{WCB}_{FA}: 347^\circ 33' + 118^\circ 52' = 466^\circ 25' * 466^\circ 25' - 360^\circ 00' = 106^\circ 25' * \text{FB}_{FA} (QB): 180^\circ 00' - 106^\circ 25' = 73^\circ 35' \implies \mathbf{S \ 73^\circ 35' \ E} 6. Line FA \to AB (Closure Check) * \text{FB}_{FA}: \mathbf{S \ 73^\circ 35' \ E} * Back Bearing \text{BB}_{FA} at A (\text{WCB}_{AF}): 106^\circ 25' + 180^\circ 00' = 286^\circ 25' * Angle \angle A: 115^\circ 10' * Calculated \text{WCB}_{AB}: 286^\circ 25' + 115^\circ 10' = 401^\circ 35' * 401^\circ 35' - 360^\circ 00' = 41^\circ 35' * Calculated \text{FB}_{AB} (QB): 41^\circ 35' \implies \mathbf{N \ 41^\circ 35' \ E} * The calculated bearing matches the given bearing, confirming the accuracy of the calculations.

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Asked by: VARRI NAVEEN
Created at: 2025-09-26 20:04:46

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# Traverse Bearing Calculation: Step-by-Step Summary This document summarizes the calculation of bearings for a closed polygonal traverse. The process uses measured interior angles and the bearing of the first line. All bearings are computed in both Whole Circle Bearing (WCB) and Quadrant Bearing (QB) formats. --- ## Summary Table | Traverse Line | Forward Bearing (QB) | |:-------------:|:-------------------:| | AB | **N 41° 35' E** (Given) | | BC | **N 9° 14' W** | | CD | **S 79° 21' W** | | DE | **S 31° 51' W** | | EF | **S 12° 27' E** | | FA | **S 73° 35' E** | --- ## Calculation Method ### General Process 1. **Start with the given bearing of AB.** 2. **At each station,** compute the Whole Circle Back Bearing (WCB BB) of the previous line. 3. **Add the measured interior angle** (clockwise, to the right) to find the new WCB. 4. **Convert the new WCB** to QB for the forward bearing of the next line. 5. **Repeat** until closure. #### Key Formulas - **Back Bearing (BB):** \[ \text{BB} = \text{FB} \pm 180^\circ \] - **Next Line WCB:** \[ \text{WCB}_{\text{next}} = \text{WCB}_{\text{back line}} + \text{Interior Angle} \] - **QB ↔ WCB Conversion:** - NE: WCB = QB - SE: WCB = 180° – QB - SW: WCB = 180° + QB - NW: WCB = 360° – QB --- ## Detailed Steps ### 1. Line AB → BC - **Given:** - \(\text{FB}_{AB} = \) **N 41° 35' E** (\(\text{WCB} = 41° 35'\)) - **BB at B:** - \(41° 35' + 180° = 221° 35'\) - **Angle at B:** - \(129° 11'\) - **WCB for BC:** - \(221° 35' + 129° 11' = 350° 46'\) - **QB for BC:** - \(360° - 350° 46' = 9° 14'\) - **FB for BC:** - **N 9° 14' W** --- ### 2. Line BC → CD - **FB for BC:** - **N 9° 14' W** (\(\text{WCB} = 350° 46'\)) - **BB at C:** - \(350° 46' - 180° = 170° 46'\) - **Angle at C:** - \(88° 35'\) - **WCB for CD:** - \(170° 46' + 88° 35' = 259° 21'\) - **QB for CD:** - \(259° 21' - 180° = 79° 21'\) - **FB for CD:** - **S 79° 21' W** --- ### 3. Line CD → DE - **FB for CD:** - **S 79° 21' W** (\(\text{WCB} = 259° 21'\)) - **BB at D:** - \(259° 21' - 180° = 79° 21'\) - **Angle at D:** - \(132° 30'\) - **WCB for DE:** - \(79° 21' + 132° 30' = 211° 51'\) - **QB for DE:** - \(211° 51' - 180° = 31° 51'\) - **FB for DE:** - **S 31° 51' W** --- ### 4. Line DE → EF - **FB for DE:** - **S 31° 51' W** (\(\text{WCB} = 211° 51'\)) - **BB at E:** - \(211° 51' - 180° = 31° 51'\) - **Angle at E:** - \(135° 42'\) - **WCB for EF:** - \(31° 51' + 135° 42' = 167° 33'\) - **QB for EF:** - \(180° - 167° 33' = 12° 27'\) - **FB for EF:** - **S 12° 27' E** --- ### 5. Line EF → FA - **FB for EF:** - **S 12° 27' E** (\(\text{WCB} = 167° 33'\)) - **BB at F:** - \(167° 33' + 180° = 347° 33'\) - **Angle at F:** - \(118° 52'\) - **WCB for FA:** - \(347° 33' + 118° 52' = 466° 25'\) - \(466° 25' - 360° = 106° 25'\) - **QB for FA:** - \(180° - 106° 25' = 73° 35'\) - **FB for FA:** - **S 73° 35' E** --- ### 6. Line FA → AB (Closure Check) - **FB for FA:** - **S 73° 35' E** (\(\text{WCB} = 106° 25'\)) - **BB at A:** - \(106° 25' + 180° = 286° 25'\) - **Angle at A:** - \(115° 10'\) - **Calculated WCB for AB:** - \(286° 25' + 115° 10' = 401° 35'\) - \(401° 35' - 360° = 41° 35'\) - **Final FB for AB:** - **N 41° 35' E** (matches original; closure confirmed) --- ## Notes - **All calculations use degrees and minutes.** - **Angles are measured clockwise (to the right) at each traverse station.** - **Closure check confirms the traverse is correctly calculated.** --- ## Diagram (Conceptual) _An actual traverse diagram is not included here, but a typical traverse would look like a polygon with labeled stations A, B, C, D, E, F, showing the direction of each bearing from point to point._ --- **Reference Illustration:** ![Polygon Traverse Example](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Closed_traverse.svg/300px-Closed_traverse.svg.png) *Alt text: Schematic diagram of a closed polygon traverse with labeled stations and lines.* --- **End of Calculation Summary**

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