'Cube' is related to 'Square' (3D solid whose face is a square) in the same way as 'Square' is related to which 1D boundary element?

Introduction / Context:
Geometric analogies often hinge on dimensional relationships. A cube is a three-dimensional solid whose faces are two-dimensional squares. We must continue the dimensional descent to find the object that stands to a square as a square stands to a cube.

Given Data / Assumptions:

• Cube (3D) → face is Square (2D).
• Square (2D) has boundary elements that are Line segments (1D).
• We keep the pattern 'figure → its (one-dimension-lower) boundary/face'.

Concept / Approach:
The mapping follows a reduction of dimension by one: 3D to 2D (faces), then 2D to 1D (edges). A square’s edges are straight line segments. Thus the consistent counterpart is 'Line' rather than a specific polygon (triangle) or a higher-dimensional entity (plane), or an even lower-dimensional vertex (point).

Step-by-Step Solution:
1) Identify relation: shape → characteristic boundary element one dimension lower.2) Cube → Square (face of the cube).3) Square → Line (edge of the square).4) Select 'Line' to preserve parallelism.

Verification / Alternative check:
A square has four sides; each side is a line segment (1D). The face of a cube is a square (2D). The analogy therefore keeps dimensional decrement consistent: 3D→2D, then 2D→1D.

Why Other Options Are Wrong:

• Plane: 2D infinite surface; not a boundary element of a square.
• Triangle: Different polygon; breaks the boundary relation.
• Point: Vertex (0D) is too far a reduction; the next direct boundary step is 1D.
• None of these: Incorrect because 'Line' is correct.

Common Pitfalls:
Jumping from faces to vertices (point) rather than edges, or choosing an unrelated polygon. Maintain the exact dimensional step-down.

Final Answer:
Line