Question

Which biconditional statement is true?

A shape is a rectangle if and only if the shape has exactly four sides and four right angles.
A shape is a trapezoid if and only if the shape has a pair of parallel sides.
A shape is a triangle if and only if the shape has three sides and three acute angles.
A shape is a square if and only if the shape has exactly four congruent sides

Answer 1

Only the rectangle statement is definitely true.

Brief justification for each:

- Rectangle: True. A rectangle is exactly a quadrilateral with four right angles, so it has exactly four sides and four right angles, and any shape with exactly four sides and four right angles is a rectangle.

- Trapezoid: Ambiguous. If "trapezoid" means “has at least one pair of parallel sides” (the inclusive definition) then the biconditional holds; if it means “has exactly one pair of parallel sides” (the exclusive definition) then it fails (a parallelogram has a pair of parallel sides but would not be a trapezoid under the exclusive definition).

- Triangle: False. Triangles need three sides but need not have three acute angles (they can be right or obtuse). A right triangle is a counterexample.

- Square: False. Every square has four congruent sides, but a quadrilateral with four congruent sides (a rhombus) need not be a square unless its angles are right; so the converse fails.