Questions: Rectangle Properties

Question 1

A parallelogram is known to have both pairs of opposite sides equal and diagonals that bisect each other. You then learn its diagonals are congruent. What can you conclude?

• A. It must be a square
• B. It must be a rectangle
• C. It must be a rhombus
• D. Nothing additional — congruent diagonals don't help classify a parallelogram

Answer: (B)

Congruent diagonals are the defining additional property that distinguishes a rectangle from a general parallelogram. The theorem works both ways: if a parallelogram has congruent diagonals, it is a rectangle. A square is a special case (also a rectangle with equal sides), but the correct classification given only this information is rectangle — not square.

Question 2

In rectangle ABCD, you draw both diagonals. Which of the following is true?

• A. The diagonals are perpendicular bisectors of each other
• B. The diagonals are congruent and bisect each other but are not necessarily perpendicular
• C. The diagonals are congruent and perpendicular but do not bisect each other
• D. The diagonals bisect each other at right angles only if the rectangle is a square

Answer: (B)

Rectangle diagonals are congruent (the property unique to rectangles among parallelograms) and bisect each other (inherited from the parallelogram). They are NOT necessarily perpendicular — perpendicular diagonals belong to rhombuses. Only when a rectangle is also a square (all sides equal) are the diagonals perpendicular. Confusing congruent diagonals with perpendicular diagonals is one of the most common errors on rectangle problems.

Question 3

Every square is a rectangle.

• T. True
• F. False

Answer: True

A rectangle is defined as a parallelogram with four right angles. A square satisfies this — it has four right angles and all four sides equal. The square is a special case of rectangle (a rectangle with equal sides), just as a rectangle is a special case of parallelogram. The hierarchy runs: square ⊂ rectangle ⊂ parallelogram ⊂ quadrilateral. Saying 'a square is not a rectangle' confuses the special case with a separate category.

Question 4

The diagonals of a rectangle are perpendicular to each other.

• T. True
• F. False

Answer: False

This is a very common misconception. Rectangle diagonals are congruent and bisect each other, but they are NOT generally perpendicular. Perpendicular diagonals are a property of rhombuses (including squares). Only when a rectangle is also a square do the diagonals happen to be perpendicular. A non-square rectangle's diagonals cross at an oblique angle — you can verify this by drawing any non-square rectangle and measuring the crossing angle.

Question 5

Why is it sufficient to prove that just one angle of a parallelogram is 90° to conclude that all four angles are 90°?

Model answer: In a parallelogram, opposite angles are equal and consecutive angles are supplementary (sum to 180°). If one angle is 90°, its opposite angle must also be 90°, and each consecutive angle must be 180° − 90° = 90°. So proving one right angle forces all four to be right angles.

This is why the definition of a rectangle only requires 'a parallelogram with right angles' — you don't need to verify all four corners. The parallelogram constraints propagate the right-angle condition around the entire figure. It also underlies the coordinate-geometry test: prove the shape is a parallelogram, then show one pair of adjacent sides is perpendicular (slopes are negative reciprocals), and you're done.