Theodolite adjustments — a theodolite is said to be in perfect adjustment when which conditions hold? Select the most comprehensive option.

Introduction / Context:
Theodolites measure horizontal and vertical angles with high precision. Proper geometry among the instrument axes is essential; otherwise, observed angles will contain systematic errors. The “perfect adjustment” conditions summarize the required orthogonality and plane constraints among axes and sighting lines.

Given Data / Assumptions:

• Definitions: vertical (rotation) axis, trunnion (horizontal) axis, and line of collimation.
• Instrument is correctly leveled so the vertical axis is truly vertical.
• Cross-hairs are properly centered and focused.

Concept / Approach:
For perfect adjustment: the vertical axis must be perpendicular to the trunnion axis so that vertical motion is truly in a vertical plane; the line of collimation must be perpendicular to the trunnion axis so that vertical circle motions do not change collimation; and as the telescope is raised or lowered, the line of sight should remain in a single vertical plane—ensuring correct vertical angle measurement and elimination of lateral sweeping errors.

Step-by-Step Solution:

Verification / Alternative check:
Reversing face observations (face left/right) and obtaining consistent angles indicate near-perfect adjustment.

Why Other Options Are Wrong:
Any single condition alone is insufficient; all must hold simultaneously to avoid systematic angular errors.

Common Pitfalls:
Assuming temporary leveling alone ensures perfect geometry; internal axis relationships also require adjustment.

Final Answer:
All of the above