---
title: "Mean angles, traverse ABCDA (2074 Asoj)"
subject: "Surveying I"
chapter: 7
key: "semester_3/surveying_i:mean-horizontal-angles-abcda-hcr-90-00-00-a-d"
source: https://www.bctengineer.com/q/semester_3/surveying_i/mean-horizontal-angles-abcda-hcr-90-00-00-a-d
site: BCT Engineering
---

# Mean angles, traverse ABCDA (2074 Asoj)

Surveying I · Chapter 7: Transit and Theodolite

- Asked in: 2074 Asoj, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/mean-horizontal-angles-abcda-hcr-90-00-00-a-d
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-7

## Question

The following observations were recorded in a theodolite traverse ABCDA. Compute the mean horizontal angles and adjust them if necessary.

| Inst. Station | Target Station | Horizontal circle reading Face Left | Face Right |
|---|---|---|---|
| A | D | 90°00′00″ | 269°59′30″ |
| | B | 204°25′40″ | 24°25′30″ |
| B | A | 90°00′00″ | 270°00′30″ |
| | C | 190°36′10″ | 10°36′00″ |
| C | B | 90°00′00″ | 269°59′50″ |
| | D | 169°08′40″ | 349°09′20″ |
| D | C | 90°00′00″ | 270°00′00″ |
| | A | 165°12′40″ | 345°12′30″ |

## Short answer

Mean angles from face left/right: A 114°25′50″, B 100°35′50″, C 79°09′05″, D 75°12′35″. Observed sum 369°23′20″, error +9°23′20″; equal correction -2°20′50″ each. Adjusted angles: A = 112°05′00″, B = 98°15′00″, C = 76°48′15″, D = 72°51′45″.

## Answer

**Assumption:** closed traverse ABCDA; the angle sum must be 360°.

**Method.** For each target the mean direction is found from the two faces:

$$
\text{Mean direction} = \frac{\text{Face L} + (\text{Face R} \pm 180^\circ)}{2}
$$

The angle at a station = mean direction of the forward station - mean direction of the back station. For a closed traverse of $n$ sides, the sum of the interior angles must be $(n-2) \times 180^\circ$; the difference is the angular error, which is distributed equally among the angles (the angles are measured with the same instrument and care).

#### Field book and mean angles
| Inst. stn | Sighted to | Face L | Face R | Mean direction | Mean angle |
|---|---|---|---|---|---|
| A | D | 90°00′00″ | 269°59′30″ | 89°59′45″ | |
| | B | 204°25′40″ | 24°25′30″ | 204°25′35″ | **114°25′50″** |
| B | A | 90°00′00″ | 270°00′30″ | 90°00′15″ | |
| | C | 190°36′10″ | 10°36′00″ | 190°36′05″ | **100°35′50″** |
| C | B | 90°00′00″ | 269°59′50″ | 89°59′55″ | |
| | D | 169°08′40″ | 349°09′20″ | 169°09′00″ | **79°09′05″** |
| D | C | 90°00′00″ | 270°00′00″ | 90°00′00″ | |
| | A | 165°12′40″ | 345°12′30″ | 165°12′35″ | **75°12′35″** |

#### Check and adjustment
Required sum = $(4-2) \times 180^\circ$ = 360°00′00″.
Observed sum = 369°23′20″.
Angular error = 369°23′20″ - 360°00′00″ = +9°23′20″.
Correction per angle = -2°20′50″ (equal for all 4 angles).

| Angle at | Mean angle | Correction | Corrected angle |
|---|---|---|---|
| A | 114°25′50″ | -2°20′50″ | **112°05′00″** |
| B | 100°35′50″ | -2°20′50″ | **98°15′00″** |
| C | 79°09′05″ | -2°20′50″ | **76°48′15″** |
| D | 75°12′35″ | -2°20′50″ | **72°51′45″** |
| Sum | 369°23′20″ | -9°23′20″ | 360°00′00″ |

**Answer:** corrected horizontal angles: A = 112°05′00″, B = 98°15′00″, C = 76°48′15″, D = 72°51′45″.

*Note on the data:* the angular error computed from the readings as printed is about 9°23′, far larger than any acceptable error for a four-sided traverse (a few tens of seconds). So there must be a recording or printing mistake in the readings (for example in 204°25′40″ or 190°36′10″). In the field, such an error would mean the angles must be re-observed rather than adjusted. The equal-correction adjustment above is shown only to demonstrate the method on the values as printed; with correct data the same steps give a small error that is adjusted in the same way.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. At A, mean directions are D = 89°59′45″ and B = 204°25′35″. What is the angle at A?
   A. 114°25′50″
   B. 114°25′35″
   C. 294°25′50″
   D. 114°25′20″
   Answer: A. 204°25′35″ - 89°59′45″ = 114°25′50″. (Key point: Angle at A)

2. The observed sum is 369°23′20″. What does this indicate?
   A. A normal error that can be adjusted
   B. An error of +9°23′20″, far too large for a four-sided traverse, so a recording mistake is likely
   C. A sign that the angle sum should be 540°
   D. An error of -9°23′20″
   Answer: B. Normal errors are a few tens of seconds; 9° suggests misprinted readings. (Key point: Size of error)

3. What is done with this error in the answer?
   A. It is added to the first angle
   B. -9°23′20″ is applied to every angle
   C. -2°20′50″ is applied to each angle (9°23′20″/4)
   D. It is ignored
   Answer: C. The equal-correction method divides the error by 4 angles and subtracts it from each. (Key point: Method on printed data)

4. What must the corrected angles total?
   A. 180°00′00″
   B. 369°23′20″
   C. 540°00′00″
   D. 360°00′00″
   Answer: D. Corrections remove the whole error, so the sum equals (4-2) x 180° = 360°. (Key point: Sum after correction)

## Close variants set in other exams

- [Mean angles, traverse ABCDA (2068 Baisakh)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-horizontal-angles-abcda-hcr-209-25-40) — Surveying I, chapter 7, asked 2×

## Related in Surveying I

- [Theodolite traverse ABCDA with missing VCR (68°30′10″)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atheodolite-traverse-abcda-hcr-vcr-68-30-10) — Surveying I, chapter 7
- [Mean angles, traverse ABCDA (2076)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-horizontal-angles-abcda-hcr-90-00-20-c-b) — Surveying I, chapter 7, asked 2×
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- [Horizontal angles from field notes ABCDA (2065)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atheodolite-field-notes-abcda-65-shrawan) — Surveying I, chapter 7
- [Mean angles and VCR, station O (reading 179°59′40″)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Amean-direction-o-abcd-first-reversed-179-59-40) — Surveying I, chapter 7, asked 3×

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