KJSEA 2025 Maths Q15 — Naming a Quadrilateral from its Properties (Rhombus)

KJSEA 2025 Grade 9 Geometry

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The Question

“A quadrilateral has the following properties: its opposite sides are equal and parallel, its diagonals bisect each other at 90 degrees, and its opposite angles are equal. What is the name of this quadrilateral?”

1

List every clue in the question

Start by separating out the three properties you are told. First, opposite sides are equal and parallel. Second, the diagonals bisect each other and cross at right angles. Third, opposite angles are equal. The trick with naming questions is that no single clue is enough on its own, so you must find the one shape that satisfies all of them at once.

2

Use the shared clues to narrow it down

Opposite sides equal and parallel, opposite angles equal, and diagonals that bisect each other are all properties of every parallelogram. That family includes the plain parallelogram, the rectangle, the square and the rhombus. So these clues alone only tell you the shape belongs to the parallelogram family. You need the diagonal angle to go further.

Parallelogram family: parallelogram, rectangle, rhombus, square\text{Parallelogram family: parallelogram, rectangle, rhombus, square}
3

Apply the deciding clue: diagonals meet at 90 degrees

The property that the diagonals cross at right angles is the special mark of a rhombus. In a plain parallelogram the diagonals bisect each other but do not meet at 90 degrees. In a rectangle the diagonals are equal in length but also do not meet at 90 degrees. Perpendicular diagonals force all four sides to be equal, which is exactly what defines a rhombus.

Diagonals bisect each other at 90∘⇒all four sides equal\text{Diagonals bisect each other at } 90^\circ \Rightarrow \text{all four sides equal}

A square also has perpendicular diagonals, but a square additionally needs right angles at its corners, which this question does not state.

4

Confirm the name

Only the rhombus fits all three properties together: opposite sides equal and parallel, opposite angles equal, and diagonals that bisect each other at 90 degrees. So the quadrilateral is a rhombus, which is option D.

Final Result

The quadrilateral is a rhombus, which is option D.

Why this method works

The reason this works is that each quadrilateral is defined by a unique combination of properties, so naming one is really a process of elimination. The shared clues place the shape in the parallelogram family, but only the diagonal condition distinguishes the members. Perpendicular diagonals are the signature of the rhombus, because when two lines that already bisect each other also meet at right angles, every side is forced to have the same length. A rectangle fails this test since its diagonals are equal but not perpendicular, and a plain parallelogram fails because its diagonals are neither equal nor perpendicular.

✓

Sketch a rhombus, draw both diagonals, and you can see they cut each other exactly in half and form four right angles at the centre, matching all three given properties.