## Step-by-Step Solution ### **Given** - ** polygon:** \(A, B, C, D, E, F\) - **Interior angles to the right at each vertex** are given. - **Bearing of \(AB\):** \(N41^\circ 35'E\) We are to **calculate the bearings for all traverse lines** using the angles provided. --- ### **Step 1: Understanding Bearings** - **Bearing** is the angle measured clockwise from the north direction. - The **bearing of the next line** can be found by **adding the internal angle (to the right)** to the **bearing of the previous line**. --- ### **Step 2: List the Angles at Each Vertex** - At \(A\): \(115^\circ 10'\) - At \(B\): \(129^\circ 11'\) - At \(C\): \(88^\circ 35'\) - At \(D\): \(132^\circ 30'\) - At \(E\): \(135^\circ 42'\) - At \(F\): \(118^\circ 52'\) --- ### **Step 3: Bearing of \(AB\) (Given)** \[ \text{Bearing of } AB = N 41^\circ 35' E = 41^\circ 35' \] --- ### **Step 4: Calculate the Bearing of \(BC\)** \[ \text{Bearing of } BC = \text{Bearing of } AB + \text{Angle at } B \] \[ \text{Bearing of } BC = 41^\circ 35' + 129^\circ 11' = 170^\circ 46' \] --- ### **Step 5: Bearing of \(CD\)** \[ \text{Bearing of } CD = \text{Bearing of } BC + \text{Angle at } C \] \[ \text{Bearing of } CD = 170^\circ 46' + 88^\circ 35' = 259^\circ 21' \] --- ### **Step 6: Bearing of \(DE\)** \[ \text{Bearing of } DE = \text{Bearing of } CD + \text{Angle at } D \] \[ \text{Bearing of } DE = 259^\circ 21' + 132^\circ 30' = 391^\circ 51' \] \[ \text{Since 391^\circ 51' > 360^\circ, \text{subtract } 360^\circ:} \] \[ \text{Bearing of } DE = 391^\circ 51' - 360^\circ = 31^\circ 51' \] --- ### **Step 7: Bearing of \(EF\)** \[ \text{Bearing of } EF = \text{Bearing of } DE + \text{Angle at } E \] \[ \text{Bearing of } EF = 31^\circ 51' + 135^\circ 42' = 167^\circ 33' \] --- ### **Step 8: Bearing of \(FA\)** \[ \text{Bearing of } FA = \text{Bearing of } EF + \text{Angle at } F \] \[ \text{Bearing of } FA = 167^\circ 33' + 118^\circ 52' = 286^\circ 25' \] --- ### **Step 9: Summary Table** | Line | Bearing (Whole Circle Bearing) | |------|-------------------------------| | AB | \(41^\circ 35'\) | | BC | \(170^\circ 46'\) | | CD | \(259^\circ 21'\) | | DE | \(31^\circ 51'\) | | EF | \(167^\circ 33'\) | | FA | \(286^\circ 25'\) | --- ## **Final Answers (Bearings of Traverse Lines):** - **AB:** \(41^\circ 35'\) - **BC:** \(170^\circ 46'\) - **CD:** \(259^\circ 21'\) - **DE:** \(31^\circ 51'\) - **EF:** \(167^\circ 33'\) - **FA:** \(286^\circ 25'\) If you need these in **reduced (quadrantal) bearing** format, let me know!

## Reduced Bearings for Traverse Lines ### **Definition** - **Reduced Bearing** is expressed in terms of north or south followed by the angle towards east or west. ### **Step-by-Step Conversion** #### **1. Convert Each Bearing** - **AB:** \(41^\circ 35'\) - **Reduced Bearing:** \(N 41^\circ 35' E\) - **BC:** \(170^\circ 46'\) - **Reduced Bearing:** \(S 9^\circ 14' W\) - (Since \(170^\circ 46' - 180^\circ = -9^\circ 14'\)) - **CD:** \(259^\circ 21'\) - **Reduced Bearing:** \(S 100^\circ 39' W\) - (Since \(259^\circ 21' - 180^\circ = 79^\circ 21'\) and is in the third quadrant) - **DE:** \(31^\circ 51'\) - **Reduced Bearing:** \(N 31^\circ 51' E\) - **EF:** \(167^\circ 33'\) - **Reduced Bearing:** \(S 12^\circ 27' W\) - (Since \(167^\circ 33' - 180^\circ = -12^\circ 27'\)) - **FA:** \(286^\circ 25'\) - **Reduced Bearing:** \(N 73^\circ 35' W\) - (Since \(286^\circ 25' - 270^\circ = 16^\circ 25'\)) ### **Summary Table of Reduced Bearings** | Line | Reduced Bearing | |------|--------------------------| | AB | \(N 41^\circ 35' E\) | | BC | \(S 9^\circ 14' W\) | | CD | \(S 100^\circ 39' W\) | | DE | \(N 31^\circ 51' E\) | | EF | \(S 12^\circ 27' W\) | | FA | \(N 73^\circ 35' W\) | ## **Final Reduced Bearings** - **AB:** \(N 41^\circ 35' E\) - **BC:** \(S 9^\circ 14' W\) - **CD:** \(S 100^\circ 39' W\) - **DE:** \(N 31^\circ 51' E\) - **EF:** \(S 12^\circ 27' W\) - **FA:** \(N 73^\circ 35' W\)