---
title: "Internal angles in quadrantal bearing system"
subject: "Surveying I"
chapter: 4
key: "semester_3/surveying_i:qb-system-internal-angles"
source: https://www.bctengineer.com/q/semester_3/surveying_i/qb-system-internal-angles
site: BCT Engineering
---

# Internal angles in quadrantal bearing system

Surveying I · Chapter 4: The Compass

- Asked in: 2072 Chaitra, 4 marks
- Video: [Calculation of Included Angle from Quadrantal Bearing](https://www.youtube.com/watch?v=VKCSo7o5rLY) — WIT Solapur - Professional Learning Community
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/qb-system-internal-angles
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-4

## Question

Explain the calculation of internal angles in the Q.B. system.

## Short answer

Interior angle is the smaller angle between back bearing of previous line and fore bearing of next line in Q.B. If same quadrant: difference; across N–S line: sum; across E–W line: 180°–sum; opposite: 180°–diff. For reflex angles, subtract from 360°. Example: BA N30°E, BC S50°E gives 100°.

## Answer

In the quadrantal bearing (Q.B.) system, the interior angle at a station is the angle between the **back line** (the back bearing of the previous line) and the **forward line** (the fore bearing of the next line). Both are written as reduced bearings, e.g. N $\theta_1$ E and S $\theta_2$ W.

#### Rules (angle $=$ smaller angle between the two lines)

| Position of the two lines | Angle |
|---|---|
| Same quadrant (e.g. both N E) | $\theta_1 \sim \theta_2$ (difference) |
| Adjacent quadrants, one on each side of the N–S line (NE & NW, or SE & SW) | $\theta_1 + \theta_2$ |
| Adjacent quadrants, one on each side of the E–W line (NE & SE, or NW & SW) | $180^\circ - (\theta_1 + \theta_2)$ |
| Opposite quadrants (NE & SW, or NW & SE) | $180^\circ - (\theta_1 \sim \theta_2)$ |

If the interior angle is the reflex angle, take $360^\circ$ minus the value found. The correct one is identified from a small sketch of the traverse; the interior angles of a closed traverse must total $(2n - 4) \times 90^\circ$.

**Example:** at B, back line BA = N $30^\circ$ E and forward line BC = S $50^\circ$ E. They lie on the east side of the meridian, one in the NE and the other in the SE quadrant (adjacent across the E–W line), so

$$
\angle ABC = 180^\circ - (30^\circ + 50^\circ) = 100^\circ
$$

(Check with WCB: BA = $30^\circ$, BC = $130^\circ$, difference $= 100^\circ$.)

## Check yourself

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

1. In QB system the interior angle at a station is between:
   A. Two fore bearings
   B. FB and BB of the same line
   C. North line and the forward line
   D. Back line (BB of previous line) and forward line (FB of next line)
   Answer: D. It is formed by the back and forward lines at the station. (Key point: Angle definition)

2. If both lines are in the same quadrant, the angle is:
   A. θ1 ~ θ2 (difference)
   B. θ1 + θ2
   C. 180° - (θ1 + θ2)
   D. 180° - (θ1 ~ θ2)
   Answer: A. Same quadrant lines differ by the difference of their reduced bearings. (Key point: Same quadrant)

3. For lines in NE and SE quadrants, the angle is:
   A. θ1 + θ2
   B. 180° - (θ1 + θ2)
   C. θ1 ~ θ2
   D. 180° + (θ1 + θ2)
   Answer: B. They lie on opposite sides of the E-W line, so the angle is 180° - (θ1 + θ2). (Key point: Across E-W line)

4. BA = N 30° E and BC = S 50° E. Angle ABC is:
   A. 80°
   B. 20°
   C. 100°
   D. 260°
   Answer: C. 180° - (30° + 50°) = 100°. (Key point: Example)

## Related in Surveying I

- [Local attraction with quadrantal bearings](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alocal-attraction-quadrantal-bearings-s11-30w) — Surveying I, chapter 4
- [Link traverse ABCDEF, right turn angles](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alink-traverse-right-turn-angles-22-17) — Surveying I, chapter 4, asked 3×
- [Open traverse ABCDEFGH, deflection angles](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aopen-traverse-deflection-angles-81-45) — Surveying I, chapter 4, asked 2×
- [Link traverse XP to TY, angles to right](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alink-traverse-xp-ty-angles-to-right) — Surveying I, chapter 4

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