If a quadrilateral is a square, then all its angles are equal.

Step-by-step solution

Idea: Equal angles fix the shape’s corners (each must be 90°) but say nothing about the sides, so a non-square rectangle breaks the converse.

1. Converse: If all the angles of a quadrilateral are equal, then it is a square.½ mark
2. Proposition — true. A square has all four sides equal and every angle 90°. So all its angles are equal (each 90°).1 mark
3. Converse. If all four angles are equal, each is 360° ÷ 4 = 90°, so the quadrilateral is a rectangle. But nothing forces its sides to be equal.½ mark
4. Counterexample: a rectangle 6 cm × 4 cm. All four angles are 90° (equal), but adjacent sides are 6 cm and 4 cm, so it is not a square. The converse is false.½ mark
5. So the proposition is true and its converse is false.½ mark

Answer to write in the exam

Common mistakes that cost marks

• Saying the converse is true because each angle is 90°. Equal angles say nothing about the sides.
• Giving a rhombus as the counterexample. A rhombus that is not a square does not have equal angles, so it does not satisfy the ‘if’ part of the converse.

How this can come in the exam

Try one yourself

Write the converse of ‘If a quadrilateral is a rhombus, then all its sides are equal’. Is the converse true?

More questions like this

• Given any ∆ABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle.
  Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown.
  Proposition: If AB = AC, then IE = IF.
• If x = y, then a + x = a + y, where x, y and a are any three numbers.
  This proposition and its converse are routinely used while solving equations.
• If a and b are perfect squares, then ab is a perfect square.
• If x = y, then x² = y².
• If x = y, then x³ = y³.

All Mathematical reasoning questions · All maths questions