Let's solve the problem step by step: --- ## **Step 1: Understanding the Problem** We are given a closed traverse ABCDEF with internal angles (measured to the right) and one known bearing (AB = N 41°55' E). We need to calculate the **bearing of each traverse line** (AB, BC, CD, DE, EF, FA). --- ## **Step 2: List the Given Data** - **Angles to the right at each point:** - A = 115°10′ - B = 129°11′ - C = 88°35′ - D = 132°30′ - E = 135°42′ - F = 118°52′ - **Known bearing:** - AB = N 41°55′ E --- ## **Step 3: Bearing Progression Formula** To find the bearing of the next line, use: > **Bearing of next line = Bearing of previous line + 180° - Interior angle at point** If the result is > 360°, subtract 360°. If the result is < °, add 360°. --- ## **Step 4: Calculate Bearings Step by Step** ### **1. Bearing of AB** Given: **N 41°55′ E** Azimuth of AB = 41°55′ (measured clockwise from North) --- ### **2. Bearing of BC** \[ \text{Azimuth of BC} = \text{Azimuth of AB} + 180° - \text{Angle at B} \] \[ = 41°55' + 180° - 129°11' \] \[ = 221°55' - 129°11' = 92°44' \] --- ### **3. Bearing of CD** \[ \text{Azimuth of CD} = \text{Azimuth of BC} + 180° - \text{Angle at C} \] \[ = 92°44' + 180° - 88°35' \] \[ = 272°44' - 88°35' = 184°09' \] --- ### **4. Bearing of DE** \[ \text{Azimuth of DE} = \text{Azimuth of CD} + 180° - \text{Angle at D} \] \[ = 184°09' + 180° - 132°30' \] \[ = 364°09' - 132°30' = 231°39' \] (Since > 360°, subtract 360° if necessary, but here it's not.) --- ### **5. Bearing of EF** \[ \text{Azimuth of EF} = \text{Azimuth of DE} + 180° - \text{Angle at E} \] \[ = 231°39' + 180° - 135°42' \] \[ = 411°39' - 135°42' = 275°57' \] (Since > 360°, subtract 360°: 275°57', so fine.) --- ### **6. Bearing of FA** \[ \text{Azimuth of FA} = \text{Azimuth of EF} + 180° - \text{Angle at F} \] \[ = 275°57' + 180° - 118°52' \] \[ = 455°57' - 118°52' = 337°05' \] (Since > 360°, subtract 360°: 337°05', so it's less than 360°, so fine.) --- ## **Step 5: Final Answers in Table** | Line | Bearing (Azimuth, from North) | |------|------------------------------| | AB | 41°55' | | BC | 92°44' | | CD | 184°09' | | DE | 231°39' | | EF | 275°57' | | FA | 337°05' | --- ### **Optional: Convert Azimuths to Quadrant Bearings** - ° to 90°: N θ E - 90° to 180°: S (180°-θ) E - 180° to 270°: S (θ-180°) W - 270° to 360°: N (360°-θ) W Let's convert: | Line | Azimuth | Quadrant Bearing | |------|---------|------------------------| | AB | 41°55' | N 41°55' E | | BC | 92°44' | S 87°16' E | | CD | 184°09' | S 4°09' W | | DE | 231°39' | S 51°39' W | | EF | 275°57' | N 84°03' W | | FA | 337°05' | N 22°55' W | --- ## **Summary Table** | Line | Azimuth | Quadrant Bearing | |------|-----------|-------------------| | AB | 41°55' | N 41°55′ E | | BC | 92°44' | S 87°16′ E | | CD | 184°09' | S 4°09′ W | | DE | 231°39' | S 51°39′ W | | EF | 275°57' | N 84°03′ W | | FA | 337°05' | N 22°55′ W | --- ### **Final Answer** > **The bearings of the traverse lines are:** > - AB: N 41°55′ E > - BC: S 87°16′ E > - CD: S 4°09′ W > - DE: S 51°39′ W > - EF: N 84°03′ W > - FA: N 22°55′ W Let me know if you need the step-by-step calculations shown for each conversion!