State whether the following statements are true or false. Justify your answers.
A quadrilateral whose diagonals are equal and bisect each other must be a square.
False. A quadrilateral with equal diagonals that bisect each other is a rectangle. For it to be a square, the diagonals must also intersect at right angles (90°).
A quadrilateral having three right angles must be a rectangle.
True. If three angles are 90° each, the fourth angle must be 360° - 270° = 90°. Thus all angles are 90°, making it a rectangle.
A quadrilateral whose diagonals bisect each other must be a parallelogram.
True. Property 4 of parallelograms states that diagonals bisect each other. Conversely, if diagonals bisect each other, the quadrilateral is a parallelogram.
A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.
False. A kite also has perpendicular diagonals. For a quadrilateral to be a rhombus, it must have all sides equal in addition to having perpendicular diagonals.
A quadrilateral in which the opposite angles are equal must be a parallelogram.
True. If opposite angles are equal, say ∠A = ∠C and ∠B = ∠D, and their sum is 360°, then adjacent angles are supplementary. This implies opposite sides are parallel, making it a parallelogram.
A quadrilateral in which all the angles are equal is a rectangle.
True. If all angles are equal and their sum is 360°, each angle is 90°. A quadrilateral with all angles 90° is a rectangle.
Isosceles trapeziums are parallelograms.
False. An isosceles trapezium has only one pair of parallel sides, while a parallelogram has two pairs of parallel sides.
Explanation
(i) The chapter establishes that when diagonals are equal and bisect each other, the quadrilateral is a rectangle (all angles are 90°). For a square, the diagonals must additionally intersect at right angles. (ii) The angle sum of a quadrilateral is 360°. If three angles are 90°, the fourth is 360° - 270° = 90°. A quadrilateral with all angles 90° is a rectangle. (iii) The chapter proves in Deduction 8 that in a parallelogram, diagonals bisect each other. The converse is also true: if diagonals bisect each other, opposite sides are equal and parallel, making it a parallelogram. (iv) A kite has perpendicular diagonals (Property 1), but not all sides are equal. A rhombus requires all sides to be equal. A square is a special case where diagonals are both perpendicular and equal. (v) Property 3 of parallelograms states that opposite angles are equal. The converse can be proved: if opposite angles are equal and sum to 360°, then adjacent angles are supplementary, which implies opposite sides are parallel. (vi) The chapter defines a rectangle as a quadrilateral with all angles equal to 90°. If all angles are equal in a quadrilateral, each angle = 360°/4 = 90°. Hence, it is a rectangle. (vii) A trapezium is defined as having at least one pair of parallel sides. An isosceles trapezium has one pair of parallel sides and non-parallel sides equal. A parallelogram requires both pairs of opposite sides to be parallel.