Triangulation adjustments: which geometric angle-sum conditions must hold for stations, triangles, and braced quadrilaterals in a properly adjusted network?

Introduction / Context:
Triangulation provides horizontal control for mapping and engineering works. Geometric constraints are used during adjustment to distribute observational errors consistently across the network and to ensure closure conditions are satisfied.

Given Data / Assumptions:

• Angles are reduced to a common datum (e.g., station mark, horizon, etc.).
• Plane-triangle approximations are applied where appropriate.
• Braced quadrilaterals are used to strengthen networks and provide checks.

Concept / Approach:
Closure and angle-sum checks safeguard the internal consistency of observed angles. Around a station, all angles subtend the full circle (360°). In plane geometry, triangle angles sum to 180°. In a braced quadrilateral, angle arrangements yield specific total sums (commonly the sum of eight selected angles equals 360°) providing a powerful redundancy check.

Step-by-Step Solution:
Station closure: confirm angles around a station total 360°.Triangle closure: enforce 180° for each triangle; distribute small misclosures by least squares or proportional rules.Braced quadrilateral checks: verify prescribed angle sums (e.g., selected eight angles) equal 360°; reconcile misclosures in adjustment.

Verification / Alternative check:
These relationships are standard and appear in triangulation specifications and adjustment textbooks as primary geometry checks.

Why Other Options Are Wrong:
Individual statements (a)–(c) are correct; therefore the most complete answer is “All the above.”

Common Pitfalls:
Neglecting to convert repeated or oriented angles to a consistent reference; omitting refraction and curvature corrections when necessary.

Final Answer:
All the above