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Internal angles in quadrantal bearing 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°.

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

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 θ1\theta_1 E and S θ2\theta_2 W.

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

Position of the two linesAngle
Same quadrant (e.g. both N E)θ1∼θ2\theta_1 \sim \theta_2 (difference)
Adjacent quadrants, one on each side of the N–S line (NE & NW, or SE & SW)θ1+θ2\theta_1 + \theta_2
Adjacent quadrants, one on each side of the E–W line (NE & SE, or NW & SW)180∘−(θ1+θ2)180^\circ - (\theta_1 + \theta_2)
Opposite quadrants (NE & SW, or NW & SE)180∘−(θ1∼θ2)180^\circ - (\theta_1 \sim \theta_2)

If the interior angle is the reflex angle, take 360∘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)×90∘(2n - 4) \times 90^\circ.

Example: at B, back line BA = N 30∘30^\circ E and forward line BC = S 50∘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

∠ABC=180∘−(30∘+50∘)=100∘\angle ABC = 180^\circ - (30^\circ + 50^\circ) = 100^\circ

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

✓ Answer checked by Claude Opus 5.5 (high effort)

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