**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 ∠CAD = 35°, then find ∠ACB.Solution:**

i. AC = 8 cm …(i) [Given]

□ABCD is a rectangle [Given]

∴ BD = AC [Diagonals of a rectangle are congruent]

∴ BD = 8 cm [From (i)]

∴ BO = 4 cm

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

∴ ∠ACB = ∠CAD [Alternate angles]

∠ACB = 35° [ ∵∠CAD = 35°]

∴ BO = 4 cm, ∠ACB = 35°

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

i. PQ = 7.5 cm [Given]

□PQRS is a rhombus. [Given]

∴ QR = PQ [Sides of a rhombus are congruent]

∴ QR = 7.5 cm

ii. ∠QPS = 75° [Given]

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

∴ 75° + ∠PQR = 180°

∴ ∠PQR = 180° – 75°

∴ ∠PQR =105°

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

∴ ∠SRQ = 75°

∴ QR = 7.5 cm, ∠PQR = 105°,

∠SRQ = 75°

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

□IJKL is a square. [Given]

∴ seg IK ⊥ seg JL [Diagonals of a square are perpendicular to each other]

∠ IMJ=90°

∠ JIL 90° ……. (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 □ABCD be the rhombus.

AC = 20 cm, BD = 21 cm

ii. In ∆AOB, ∠AOB = 90° [Diagonals of a rhombus are prependicular to each other]

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

iii. Perimeter of □ABCD

= 4 x AB = 4 x 14.5 = 58 cm

∴ 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 ‘true’ or ‘false’.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.