--- title: "Angle misclosure, closed link traverse AT1-ET2" subject: "Surveying I" chapter: 4 key: "semester_3/surveying_i:closed-link-traverse-at1-et2" source: https://www.bctengineer.com/q/semester_3/surveying_i/closed-link-traverse-at1-et2 site: BCT Engineering --- # Angle misclosure, closed link traverse AT1-ET2 Surveying I · Chapter 4: The Compass - Asked in: 2058 Chaitra, 10 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/closed-link-traverse-at1-et2 - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-4 ## Question Compute the angle misclosure and adjusted bearings in the following closed link traverse. Adjusted bearing AT1 = 329°09′21″ [?]. Adjusted bearing ET2 = 105°36′08″. Angles (angles to the right): A = 282°17′18″, B = 266°48′13″, C = 89°16′53″, D = 96°36′05″, E = 291°27′38″. ## Short answer Angle misclosure 40'' (computed ET2 105°35'28''). Correction +8'' per angle, giving adjusted bearings: T1A 339°09'21'', AB 81°26'47'', BC 168°15'08'', CD 77°32'09'', DE 354°08'22'', ET2 105°36'08''. ## Answer **Reading of the data.** The first bearing is printed as 329°09′21″ [?]. If 329°09′21″ is taken as the bearing of the line from T1 to A, the computed bearing of ET2 comes out as 95°35′28″, a misclosure of about 10°00′40″, which is impossible for angles measured to the second. The value consistent with the other data is **339°09′21″** for the line from T1 to A (its reverse bearing, from A to T1, is 159°09′21″). That value is used below. The angles are to the right, so the bearing of each forward line = bearing of the previous line + 180° + angle at the station. **Rule (angles to the right):** $\text{FB of next line} = \text{FB of previous line} + 180^\circ + \text{angle to the right}$ (add or subtract $360^\circ$ to bring the result between $0^\circ$ and $360^\circ$). #### Step 1: Bearings computed from the observed angles | Line | Working | Computed FB | |---|---|---| | T1-A | given | 339°09′21″ | | A-B | 339°09′21″ + 180° + 282°17′18″ − 720° | 81°26′39″ | | B-C | 81°26′39″ + 180° + 266°48′13″ − 360° | 168°14′52″ | | C-D | 168°14′52″ + 180° + 89°16′53″ − 360° | 77°31′45″ | | D-E | 77°31′45″ + 180° + 96°36′05″ | 354°07′50″ | | E-T2 | 354°07′50″ + 180° + 291°27′38″ − 720° | 105°35′28″ | Computed FB of E-T2 = 105°35′28″; given (known) FB of E-T2 = 105°36′08″. #### Step 2: Angular misclosure $$ \text{Misclosure} = \text{given} - \text{computed} = 0^\circ 00'40'' $$ The computed bearing is 40″ too small. This is only 8″ per angle for 5 angles, which is acceptable for a 20″ theodolite (the permissible value $C\sqrt{n}$ with $C = 20''$ is about $45''$). The error is distributed equally among the 5 angles. #### Step 3: Adjustment Total correction = +0°00′40″ (added to the computed bearing of the last line). It is distributed equally over the 5 observed angles: +0°00′40″ / 5 = +0°00′08″ per angle. The correction to a line is +0°00′08″ multiplied by the number of angles used to reach it. | Line | Computed FB | Correction | Corrected FB | Corrected BB | |---|---|---|---|---| | T1-A | 339°09′21″ | 0 | 339°09′21″ | 159°09′21″ | | A-B | 81°26′39″ | +0°00′08″ | 81°26′47″ | 261°26′47″ | | B-C | 168°14′52″ | +0°00′16″ | 168°15′08″ | 348°15′08″ | | C-D | 77°31′45″ | +0°00′24″ | 77°32′09″ | 257°32′09″ | | D-E | 354°07′50″ | +0°00′32″ | 354°08′22″ | 174°08′22″ | | E-T2 | 105°35′28″ | +0°00′40″ | 105°36′08″ | 285°36′08″ | Check: the corrected FB of the last line equals the given value 105°36′08″. **Answer:** angle misclosure = 40″ (computed ET2 = 105°35′28″, adjusted ET2 = 105°36′08″). Adjustment = +8″ per angle (bearing corrections +8″, +16″, +24″, +32″, +40″). Adjusted bearings: T1A 339°09′21″, AB 81°26′47″, BC 168°15′08″, CD 77°32′09″, DE 354°08′22″, ET2 105°36′08″. *Note:* the adjusted angles are the observed angles plus 8″ each; the correction is added to each angle to the right because the computed bearing is too small. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For angles measured to the right, how is the bearing of the next line computed? A. Previous bearing − 180° − angle B. Previous bearing + 180° + angle (reduced to 0°–360°) C. Previous bearing + angle only D. Previous bearing − angle Answer: B. FB of next line = FB of previous line + 180° + angle to the right, adding or subtracting 360° as needed. (Key point: Angle-right rule) 2. The computed bearing of ET2 is 105°35′28″ and the given value is 105°36′08″. What is the angular misclosure (given − computed)? A. −40″ B. +40″ C. +80″ D. +8″ Answer: B. 105°36′08″ − 105°35′28″ = +40″. (Key point: Misclosure) 3. How is the 40″ error distributed among the 5 angles, and what is the bearing correction for the third line (B-C)? A. 8″ per angle; cumulative correction +16″ at line B-C B. 8″ per angle; +24″ at line B-C C. 40″ at each line D. 5″ per angle; +10″ at line B-C Answer: A. 40″/5 = 8″ per angle. The correction to a line is 8″ times the number of angles used to reach it; B-C is reached by 2 angles (A and B), so +16″. (Key point: Distributing the error) 4. Why is a 40″ misclosure for 5 angles acceptable? A. It is below the permissible C√n = 20″ × √5 ≈ 45″ B. Any misclosure under 1′ is acceptable C. It equals zero after rounding D. It is only acceptable when bearings are quadrantal Answer: A. The permissible misclosure is C√n; with C = 20″ and n = 5 it is about 45″, so 40″ is acceptable. 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