(i) A quadrilateral whose diagonals are equal and bisect each other need not be a square. In the figure, diagonals AC and DB are equal and bisect each other. Such a quadrilateral is always a rectangle. ∴ The given statement is false. (ii) Let ABCD be a quadrilateral having three right angles at A, D, and C. We have ∠A + ∠B + ∠C + ∠D = 360°. ⇒ 90° + ∠B + 90° + 90° = 360° ⇒ ∠B = 360° – 270° ⇒ ∠B = 90°. ∴ Each angle of ABCD is 90°. ∴ Given quadrilateral is a rectangle. ∴ The given statement is true. (iii) In the quadrilateral ABCD, the diagonals AC and BD bisect each other. Here, ΔAOD and ΔCOB are congruent. ∴ ∠1 = ∠2 ∴ BC is parallel to AD. Also, ΔAOB and ΔCOD are congruent. ∴ ∠3 = ∠4 ∴ AB is parallel to DC. Since opposite sides of ABCD are parallel, it must be a parallelogram. ∴ The given statement is true. (iv) Let ABCD be a quadrilateral whose diagonals AC and BD are perpendicular to each other. This quadrilateral may not be a rhombus, because the diagonals AC and BC may not bisect each other. ∴ The given statement is false. (v) Let ABCD be a quadrilateral in which ∠1 = ∠3 and ∠2 = ∠4. We have, ∠1 + ∠2 + ∠3 + ∠4 = 360°. ⇒ ∠1 + ∠2 + ∠1 + ∠2 = 360° ⇒ 2(∠1 + ∠2) = 360° ⇒ ∠1 + ∠2 = 180° AB is a transversal of lines AD and BC, and the sum of internal angles ∠1 and ∠2 on the same side is 180°. ∴ Lines AD and BC are parallel. Again, ∠1 + ∠2 + ∠3 + ∠4 = 360° ⇒ ∠3 + ∠2 + ∠3 + ∠2 = 360° ⇒ 2(∠2 + ∠3) = 360° ⇒ ∠2 + ∠3 = 180° BC is a transversal of lines AB and DC, and the sum of internal angles ∠2 and ∠3 on the same sides is 180°. ∴ Lines AB and DC are parallel. ∴ Opposite sides of quadrilateral ABCD are parallel. ∴ ABCD is a parallelogram. ∴ The given statement is true. (vi) Let ABCD be a quadrilateral, where ∠1, ∠2, ∠3, and ∠4 are all equal. We have ∠1 + ∠2 + ∠3 + ∠4 = 360° ∴ ∠1 + ∠1 + ∠1 + ∠1 = 360° ⇒ 4∠1 = 360° ⇒ ∠1 = 90° ∴ ∠2 = 90°, ∠3 = 90°, ∠4 = 90° We have, ∠5 + ∠6 = 90° and ∠6 + 90° + ∠8 = 180° ⇒ ∠5 + ∠6 = ∠6 + ∠8 ⇒ ∠5 = ∠8 Also, ∠7 + 90° + ∠5 = 180° ⇒ ∠7 + ∠5 = 90° ⇒ ∠5 + ∠6 = ∠7 + ∠5 ⇒ ∠6 = ∠7 In ΔDAB and ΔBCD, we have ∠5 = ∠8, ∠7 = ∠6, and side BD is common. ∴ By the ASA condition, ΔDAB and ΔBCD are congruent. ∴ DA = BC and AB = CD ∴ Opposite sides of ABCD are equal. ∴ ABCD is a rectangle. ∴ The given statement is true (vii) An isosceles trapezium ABCD can not be a parallelogram because it has two non-parallel equal lines AD and BC. ∴ The given statement is false.