Special parallelograms · 4 marks

Review all the properties of a rhombus/rectangle/square that you proved earlier. Formulate a converse of each. Decide if the converse is true. There are many possibilities here!

Answer: Sample results. True converses: a quadrilateral whose diagonals bisect each other at right angles is a rhombus; whose diagonals bisect each other and are equal is a rectangle; whose diagonals are equal and bisect each other at right angles is a square; whose diagonals bisect all four angles is a rhombus; with all angles 90° is a rectangle. False converses: “equal diagonals ⇒ rectangle” (isosceles trapezium); “perpendicular diagonals ⇒ rhombus” (kite); “equal and perpendicular diagonals ⇒ square” (a kite with equal diagonals).

Step-by-step solution

Idea: Write each property as “If ABCD is a rhombus (rectangle, square), then …”, swap the two parts, and test: prove it with congruent triangles or find one shape that breaks it.

Rhombus. (1) Diagonals bisect each other at right angles. Converse “diagonals bisect each other at 90° ⇒ rhombus”: true (it is a parallelogram, and SAS on the right angles makes adjacent sides equal). (2) Each diagonal bisects the angles at its ends. Converse “both diagonals bisect the angles ⇒ rhombus”: true (ASA twice gives all sides equal). “Diagonals perpendicular ⇒ rhombus”: false (a kite).1½ marks
Rectangle. (1) Diagonals are equal and bisect each other. Converse “diagonals equal and bisecting each other ⇒ rectangle”: true. “Diagonals equal ⇒ rectangle”: false (isosceles trapezium). (2) All angles are 90°. Converse “all angles 90° ⇒ rectangle”: true (co-interior angles give both pairs of sides parallel).1½ marks
Square. Diagonals are equal and are perpendicular bisectors of each other. Converse: true. But “equal and perpendicular diagonals ⇒ square” is false: the kite P(0, 4), Q(2, 1), R(0, 0), S(−2, 1) has PR = QS = 4 and PR ⟂ QS.1 mark

Many converses are true: diagonals bisecting each other at right angles ⇒ rhombus; diagonals equal and bisecting each other ⇒ rectangle; diagonals equal and perpendicular bisectors of each other ⇒ square; both diagonals bisecting the angles ⇒ rhombus; all angles 90° ⇒ rectangle. Converses about diagonals that drop “bisect each other” are false: equal diagonals ⇒ rectangle, perpendicular diagonals ⇒ rhombus, equal perpendicular diagonals ⇒ square.

Answer to write in the exam

Rhombus: diagonals ⟂ bisectors of each other ⇒ rhombus (true); diagonals bisect all angles ⇒ rhombus (true); diagonals ⟂ only ⇒ rhombus (false: kite)

Rectangle: diagonals equal and bisecting each other ⇒ rectangle (true); diagonals equal only (false: isosceles trapezium); all angles 90° ⇒ rectangle (true)

Square: diagonals equal, ⟂ and bisecting each other ⇒ square (true); equal and ⟂ only (false: kite)

∴ Converses about the diagonals are true when they keep “the diagonals bisect each other”; dropping it makes them false (kite, isosceles trapezium).

Common mistakes that cost marks

Assuming every converse is true because the property is true.
Dropping a condition when writing the converse (for example forgetting “bisect each other”).
Giving “false” without a counterexample, or “true” without a proof.

How this can come in the exam

MCQ (1 mark)

Which statement is true?

A quadrilateral with perpendicular diagonals is a rhombus
A quadrilateral with equal diagonals is a rectangle
A quadrilateral whose diagonals bisect each other and are equal is a rectangle
A quadrilateral with equal and perpendicular diagonals is a square

Show answer

(C) A quadrilateral whose diagonals bisect each other and are equal is a rectangle
Diagonals bisecting each other make a parallelogram, and a parallelogram with equal diagonals is a rectangle. The others fail for a kite or an isosceles trapezium.

Try one yourself

State the converse of “every angle of a rectangle is 90°” and decide if it is true.