Let PQRS be a parallelogram where the length of side PQ is 15 centimeters (cm). Which of the following statements...

1. TRANSLATE the problem setup

• Given information:
  • PQRS is already a parallelogram
  • Side \(\mathrm{PQ = 15\text{ cm}}\)
• What we need: Sufficient condition to prove it's a rhombus

2. INFER what makes a rhombus special

• A rhombus is a parallelogram where all four sides are equal
• Since PQRS is a parallelogram, we know opposite sides are equal: \(\mathrm{PQ = RS}\) and \(\mathrm{QR = SP}\)
• We already know \(\mathrm{PQ = 15\text{ cm}}\), so \(\mathrm{RS = 15\text{ cm}}\) too
• The key insight: We need to prove the adjacent sides (\(\mathrm{QR}\) and \(\mathrm{SP}\)) are also 15 cm

3. INFER the requirements for each answer choice

Let's test what each condition tells us:

Choice A: Angle PQR is a right angle

• This makes the parallelogram a rectangle
• Rectangles can have right angles but different side lengths (like \(\mathrm{15\text{ cm} \times 10\text{ cm}}\))
• Not sufficient to prove all sides are equal

Choice B: Diagonals PR and QS are congruent

• Congruent diagonals are a property of rectangles, not rhombi
• In rhombi, diagonals are perpendicular but not necessarily equal in length
• Not sufficient to prove all sides are equal

Choice C: QR = 15 cm

• If \(\mathrm{QR = 15\text{ cm}}\), then its opposite side \(\mathrm{SP = 15\text{ cm}}\) (parallelogram property)
• Now all four sides equal 15 cm: \(\mathrm{PQ = QR = RS = SP = 15\text{ cm}}\)
• This is sufficient to prove PQRS is a rhombus!

Choice D: RS = 15 cm

• We already knew this from parallelogram properties (\(\mathrm{RS = PQ = 15\text{ cm}}\))
• This gives us no new information about the adjacent sides \(\mathrm{QR}\) and \(\mathrm{SP}\)
• Not sufficient

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the properties of rectangles and rhombi, thinking that right angles or congruent diagonals are what make a rhombus special.

They might think: "A rhombus looks like a 'diamond' shape, so it must have right angles" or "All special parallelograms have congruent diagonals." This leads them to incorrectly select Choice A (right angle) or Choice B (congruent diagonals).

Second Most Common Error:

Incomplete INFER reasoning: Students recognize they need equal sides but don't fully analyze what information they already have versus what they need.

They might select Choice D (RS = 15 cm) thinking this provides new information, when they should realize this is already implied by the parallelogram properties. This happens when they don't systematically think through what the parallelogram properties already tell them.

The Bottom Line:

This problem tests whether students can distinguish between the defining properties of different special parallelograms and systematically determine what additional information is actually needed beyond what's already given.