The problem provided involves a traverse with three sides, where various aspects need to be computed, including the angular misclosure, adjusted angles, bearings, linear misclosure, and coordinates for two unknown points. Here’s a step-by-step breakdown for each part:

Part a) Angular Misclosure (in sexagesimal system)

1. Compute the angular misclosure:

   • The sum of internal angles in a triangle should be 180° (in a planar system).
   • Sum the given angles: $56^\circ 59' 32" + 80^\circ 42' 28" + 42^\circ 17' 45"$
   • Convert to decimal degrees and compute the sum.

2. Check if the misclosure is within acceptable limits:

   • Calculate the difference between 180° and the sum of angles from the first step.
   • If the misclosure is less than ±20", it's considered acceptable.

Part b) Adjusted Internal Angles

1. Distribute the angular misclosure:

   • The misclosure should be equally distributed across the three angles.
   • Adjust each angle accordingly by adding or subtracting the appropriate amount.

2. Recompute the angular misclosure:

   • After adjusting, ensure that the sum of angles equals exactly 180°.

Part c) Adjusted Bearings of Each Traverse Line

1. Calculate the bearings:
   • Start with the given bearing from point A to B (10°30'00").
   • Using the adjusted internal angles, calculate the bearings for the remaining two legs (B to C, and C to A).

Part d) Linear Misclosure and Precision

1. Compute the horizontal vector components:

   • For each line (AB, BC, CA), calculate the change in Easting (ΔE) and Northing (ΔN) using the distances and bearings.

2. Calculate the linear misclosure:

   • Use the vector components to find the misclosure by determining how far the computed endpoint is from the known point.

3. Precision of the traverse:

   • The precision is calculated by the ratio of the total perimeter to the linear misclosure.
   • A precision better than 1:15,000 is acceptable.

Part e) Coordinates for Mark B and Mark C

1. Using the known coordinates of A (100.000E, 100.000N):
   • Use the bearings and distances to calculate the coordinates for points B and C.

Let’s go through each calculation in detail. Would you like to start with the angular misclosure computation, or do you prefer to move on to another part of the problem? Let me know if you want more detailed steps on any specific part!

Expansion Questions:

1. What is the formula to convert degrees, minutes, and seconds into decimal degrees?
2. How do you calculate the Easting and Northing components using distances and bearings?
3. What is the significance of linear misclosure in surveying?
4. How is traverse precision typically calculated?
5. Why is it important to adjust the angular misclosure evenly across the angles?

Tip:

When calculating bearings and distances, it’s crucial to use accurate trigonometric functions to avoid compounding small errors in large-scale surveying tasks.