A regular triangular prism consists of $2$ congruent equilateral triangular bases and $3$ identical rectangular lateral faces. Which of the following statements regarding its 2D net unfoldings and spatial geometry is/are TRUE?
Options:
(A) The solid has exactly $6$ vertices, $9$ edges, and $5$ faces.
(B) A valid net can be formed by three rectangles joined in a row, with the two triangular bases attached to opposite sides of the middle rectangle.
(C) A valid net can have both triangular bases attached to the same edge of the same rectangular face.
(D) The sum of all internal dihedral angles formed by its adjacent faces is greater than $360^\circ$.
Answer Key:
A, B, D
Step-by-Step Solution & Explanation
**Method 1: Spatial Prism Geometry**
- (A) True: A triangular prism has 6 vertices, 9 edges, and 5 faces ($V - E + F = 6 - 9 + 5 = 2$).
- (B) True: Three lateral rectangles joined edge-to-edge in a strip with triangular flaps on opposite sides of the middle rectangle fold into a complete closed prism.
- (C) False: Attaching both triangular bases to the same edge of a single rectangle would cause them to overlap and collide upon folding, failing to close both opposite ends.
- (D) True: The 3 lateral-to-lateral dihedral angles are each $60^\circ$ ($3 \times 60^\circ = 180^\circ$) and the 6 base-to-lateral dihedral angles are each $90^\circ$ ($6 \times 90^\circ = 540^\circ$). Total sum $= 180^\circ + 540^\circ = 720^\circ > 360^\circ$.
Therefore, options **A**, **B**, and **D** are true.