**Chapter 5 Quadrilaterals Practice Set 5.3**

Chapter 5 Quadrilaterals Practice Set 5.3

**Question 1.Diagonals of a rectangle ABCD intersect at point O. If AC = 8 cm, then find BO and if âˆ CAD = 35Â°, then find âˆ ACB.Solution:**

i. AC = 8 cm â€¦(i) [Given]

â–¡ABCD is a rectangle [Given]

âˆ´ BD = AC [Diagonals of a rectangle are congruent]

âˆ´ BD = 8 cm [From (i)]

âˆ´ BO = 4 cm

ii. side AD || side BC and seg AC is their transversal. [Opposite sides of a rectangle are parallel]

âˆ´ âˆ ACB = âˆ CAD [Alternate angles]

âˆ ACB = 35Â° [ âˆµâˆ CAD = 35Â°]

âˆ´ BO = 4 cm, âˆ ACB = 35Â°

**Question 2.In a rhombus PQRS, if PQ = 7.5 cm, then find QR. If âˆ QPS 75Â°, then find the measures of âˆ PQR and âˆ SRQ.Solution:**

i. PQ = 7.5 cm [Given]

â–¡PQRS is a rhombus. [Given]

âˆ´ QR = PQ [Sides of a rhombus are congruent]

âˆ´ QR = 7.5 cm

ii. âˆ QPS = 75Â° [Given]

âˆ QPS + âˆ PQR = 180Â° [Adjacent angles of a rhombus are supplementary]

âˆ´ 75Â° + âˆ PQR = 180Â°

âˆ´ âˆ PQR = 180Â° â€“ 75Â°

âˆ´ âˆ PQR =105Â°

iii. âˆ SRQ = âˆ QPS [Opposite angles of a rhombus]

âˆ´ âˆ SRQ = 75Â°

âˆ´ QR = 7.5 cm, âˆ PQR = 105Â°,

âˆ SRQ = 75Â°

**Question 3.Diagonals of a square IJKL intersects at point M. Find the measures of âˆ IMJ, âˆ JIK and âˆ LJK.Solution:**

â–¡IJKL is a square. [Given]

âˆ´ seg IK âŠ¥ seg JL [Diagonals of a square are perpendicular to each other]

âˆ IMJ=90Â°

âˆ JIL 90Â° â€¦â€¦. (i) [Angle of a square]

**Question 4.Diagonals of a rhombus are 20 cm and 21 cm respectively, then find the side of rhombus and its Perimeter.Solution:**

i. Let â–¡ABCD be the rhombus.

AC = 20 cm, BD = 21 cm

ii. In âˆ†AOB, âˆ AOB = 90Â° [Diagonals of a rhombus are prependicular to each other]

âˆ´ AB^{2} = AO^{2} + BO^{2} [Pythagoras theorem]

iii. Perimeter of â–¡ABCD

= 4 x AB = 4 x 14.5 = 58 cm

âˆ´ The side and perimeter of the rhombus are 14.5 cm and 58 cm respectively.

**Question 5.State with reasons whether the following statements are â€˜trueâ€™ or â€˜falseâ€™.i. Every parallelogram is a rhombus.ii. Every rhombus is a rectangle,iii. Every rectangle is a parallelogram.iv. Every square is a rectangle,v. Every square is a rhombus.vi. Every parallelogram is a rectangle.Answer:**

i. False.

All the sides of a rhombus are congruent, while the opposite sides of a parallelogram are congruent.

ii. False.

All the angles of a rectangle are congruent, while the opposite angles of a rhombus are congruent.

iii. True.

The opposite sides of a parallelogram are parallel and congruent. Also, its opposite angles are congruent.

The opposite sides of a rectangle are parallel and congruent. Also, all its angles are congruent.

iv. True.

The opposite sides of a rectangle are parallel and congruent. Also, all its angles are congruent.

All the sides of a square are parallel and congruent. Also, all its angles are congruent.

v. True.

All the sides of a rhombus are congruent. Also, its diagonals are perpendicular bisectors of each other.

All the sides of a square are congruent. Also, its diagonals are perpendicular bisectors of each other.

vi. False.

All the angles of a rectangle are congruent, while the opposite angles of a parallelogram are congruent.