Question:

Given Information: ​Starting Bearing: Bearing of AB = N 41°35' E (which is 41°35' from North clockwise, or an azimuth of 41°35'). ​Interior Angles: ​Angle at B = 129°11' ​Angle at C = 88°35' ​Angle at D = 132°30' ​Angle at E = 135°42' ​Angle at F = 118°52' ​Angle at A = 115°10' ​Key Principles: ​Forward Bearing (FB) and Back Bearing (BB): BB = FB \pm 180^\circ. If FB < 180^\circ, then BB = FB + 180^\circ. If FB > 180^\circ, then BB = FB - 180^\circ. ​Bearing of Next Line: Bearing of next line = Back Bearing of previous line \pm Interior Angle. ​If the traverse is run clockwise (like ABCDEF), then Bearing of next line = Back Bearing of previous line + Interior Angle. ​If the result is > 360^\circ, subtract 360^\circ. ​If the result is < 0^\circ, add 360^\circ. ​Correction for Left/Right Turns: A more general rule is: Bearing of next line = Back Bearing of previous line + Interior Angle (if turning right) or - Interior Angle (if turning left). Since this is a closed polygon, we follow the direction of traverse. For an interior angle with a clockwise traverse, the formula is generally: Bearing of Next Line = Bearing of Previous Line + Interior Angle - 180° (this is usually for azimuths, and needs careful application). ​Let's use a simpler, more robust method for azimuths: Azimuth of next line = Azimuth of previous line (Back Bearing) + Interior Angle - 180° (if the traverse is clockwise, which this one is). If the result is outside 0-360^\circ, adjust by adding/subtracting 360^\circ. ​Let's convert the initial bearing to an azimuth (whole circle bearing) first. Bearing of AB = N 41°35' E. This means it's in the first quadrant. Azimuth of AB = 41°35' ​Calculations: ​1. Line AB: ​Azimuth of AB = 41°35' ​2. Line BC: ​Back Azimuth of AB = 41°35' + 180° = 221°35' ​Azimuth of BC = Back Azimuth of AB - Angle at B + 360° (if needed) ​Since the traverse is clockwise, we use: Azimuth of BC = Back Azimuth of AB + Interior Angle at B - 180° ​Azimuth of BC = 221°35' + 129°11' - 180° ​Azimuth of BC = 350°46' - 180° = 170°46' ​Bearing of BC: Since 170^\circ46' is between 90^\circ and 180^\circ, it's in the SE quadrant. 180^\circ - 170^\circ46' = 9^\circ14'. So, S 9°14' E. ​3. Line CD: ​Back Azimuth of BC = 170°46' + 180° = 350°46' ​Azimuth of CD = Back Azimuth of BC + Angle at C - 180° ​Azimuth of CD = 350°46' + 88°35' - 180° ​Azimuth of CD = 439°21' - 180° = 259°21' ​Bearing of CD: Since 259^\circ21' is between 180^\circ and 270^\circ, it's in the SW quadrant. 259^\circ21' - 180^\circ = 79^\circ21'. So, S 79°21' W. ​4. Line DE: ​Back Azimuth of CD = 259°21' - 180° = 79°21' ​Azimuth of DE = Back Azimuth of CD + Angle at D - 180° ​Azimuth of DE = 79°21' + 132°30' - 180° ​Azimuth of DE = 211°51' - 180° = 31°51' ​Bearing of DE: Since 31^\circ51' is between 0^\circ and 90^\circ, it's in the NE quadrant. So, N 31°51' E. ​5. Line EF: ​Back Azimuth of DE = 31°51' + 180° = 211°51' ​Azimuth of EF = Back Azimuth of DE + Angle at E - 180° ​Azimuth of EF = 211°51' + 135°42' - 180° ​Azimuth of EF = 347°33' - 180° = 167°33' ​Bearing of EF: Since 167^\circ33' is between 90^\circ and 180^\circ, it's in the SE quadrant. 180^\circ - 167^\circ33' = 12^\circ27'. So, S 12°27' E. ​6. Line FA: ​Back Azimuth of EF = 167°33' + 180° = 347°33' ​Azimuth of FA = Back Azimuth of EF + Angle at F - 180° ​Azimuth of FA = 347°33' + 118°52' - 180° ​Azimuth of FA = 466°25' - 180° = 286°25' ​Bearing of FA: Since 286^\circ25' is between 270^\circ and 360^\circ, it's in the NW quadrant. 360^\circ - 286^\circ25' = 73^\circ35'. So, N 73°35' W. ​7. Line AB (Closure Check): ​Back Azimuth of FA = 286°25' - 180° = 106°25' ​Azimuth of AB = Back Azimuth of FA + Angle at A - 180° ​Azimuth of AB = 106°25' + 115°10' - 180° ​Azimuth of AB = 221°35' - 180° = 41°35' ​This matches the initial Azimuth of AB (41°35'), so the calculations are consistent! ​Summary of Bearings: ​Bearing of AB: N 41°35' E ​Bearing of BC: S 9°14' E ​Bearing of CD: S 79°21' W ​Bearing of DE: N 31°51' E ​Bearing of EF: S 12°27' E ​Bearing of FA: N 73°35' W