Question

which of the given conditional statements has no counterexample?
if a quadrilateral has two pairs of congruent sides, then it is a parallelogram.
if a quadrilateral has all angles congruent, then it is a square.
if a quadrilateral has diagonals that are perpendicular bisectors of each other, then it is a rhombus.
if a quadrilateral has at least two angles that are congruent, then it is a trapezoid.

Explanation:

Step1: Analyze each option

• Option 1: A quadrilateral with two pairs of congruent sides could be a parallelogram, but also a kite (which is not a parallelogram in all cases), so counterexample exists.
• Option 2: A quadrilateral with all angles congruent (each \(90^\circ\)) must be a rectangle (a type of parallelogram), so no counterexample.
• Option 3: A quadrilateral with perpendicular diagonals could be a rhombus or a kite, but a kite with unequal adjacent sides is not a rhombus, so counterexample exists.
• Option 4: A quadrilateral with two congruent angles could be many shapes, not necessarily a trapezoid (e.g., a parallelogram with two congruent angles but not a trapezoid in some definitions), so counterexample exists.

Step2: Identify the correct option

From the analysis, the statement "If a quadrilateral has all angles congruent, then it is a parallelogram" has no counterexample.

Answer:

The option (assuming the second option in the list) with the statement "If a quadrilateral has all angles congruent, then it is a parallelogram"