Adjustment of a geodetic quadrilateral: If both diagonals are observed, which independent condition equations are applied during figure adjustment?

Introduction / Context:
In geodetic surveying, a quadrilateral figure is strengthened by observations on sides, angles, and (optionally) diagonals. The number and nature of condition equations govern the least-squares adjustment and the internal consistency of the measurements.

Given Data / Assumptions:

• Quadrilateral with all station angles observed.
• Both diagonals measured.
• Objective: determine the appropriate set of independent conditions.

Concept / Approach:
Closure conditions enforce geometric consistency. In a quadrilateral, angle conditions capture angular closures around the figure (including spherical excess as applicable), while side conditions enforce that diagonal lengths computed from adjacent triangles agree with the measured diagonals. Having both diagonals available yields two independent side conditions (one for each diagonal) in addition to angular closure relations.

Step-by-Step Solution:
Establish angular closure relations for the figure → two independent angle equations.Form length consistency for each diagonal (computed from either triangle pair sharing the diagonal) → two independent side equations.Total independent condition equations = 4: two angles + two sides.

Verification / Alternative check:
Textbook treatments show that a quadrilateral without diagonals has two angle conditions; adding one diagonal adds one length condition; adding the second diagonal adds the second length condition.

Why Other Options Are Wrong:
One angle + three sides or three angles + one side miscount the independent relations for the observed configuration.

Common Pitfalls:
Confusing the number of measured quantities with the number of independent conditions; independence must be checked to avoid redundant equations in adjustment.

Final Answer:
Two angle equations and two side (length) equations.