Question

which statement describes a parallelogram that must be a square?

a parallelogram with all sides congruent.

a parallelogram with a right angle and opposites sides that are congruent.

a parallelogram with opposite sides that are congruent and diagonals that bisect the angles

a parallelogram with a right angle and diagonals that bisect the angles.

Explanation:

Step1: Analyze Option A

A parallelogram with all sides congruent is a rhombus. A rhombus doesn't necessarily have right angles, so it's not always a square.

Step2: Analyze Option B

A parallelogram with a right angle and opposite sides congruent is a rectangle (since a parallelogram with a right angle is a rectangle, and opposite sides congruent is always true for parallelograms). A rectangle doesn't necessarily have all sides congruent, so it's not always a square.

Step3: Analyze Option C

A parallelogram with opposite sides congruent is just a general parallelogram (opposite sides are always congruent in parallelograms), and diagonals that bisect the angles is a property of a rhombus. But a rhombus doesn't necessarily have right angles, so this is a rhombus, not necessarily a square.

Step4: Analyze Option D

A parallelogram with a right angle is a rectangle. If the diagonals of a rectangle bisect the angles, then each triangle formed by a diagonal is an isosceles right triangle, so all sides are congruent. So a rectangle with diagonals that bisect the angles is a square.

Answer:

D. A parallelogram with a right angle and diagonals that bisect the angles.