**Class 8 Maths Prachi Publication Chapter 7**Understanding Shapes.

In this chapter, we have been given the solution of Assignment 6.1, 6.2 and 6.3 in a simple and easy way by Study Circle.

**Class 8 Maths Prachi Publication Chapter 7**Understanding Shapes and Exercises also contains solutions for Section A and Section B. These types of solutions provided by Study Circle will be very helpful for the students and will be of great help to them.

**Class – 8 Maths (Prachi Publication) chapter – 7**

**Contents**hide

**ASSIGNMENT 7.1**

**Question 1- Three angles of a quadrilateral are 100°, 50° and 50° respectively Find the measure of fourth angle. **

**(a) 140° (b) 160° (c) 120° (d) 100° **

Solution:–

** **

**(iii) The four angles of a quadrilateral are in the ratio 1: 3: 7: 9. which of the following is not the angle of this quadrilateral ? **

**(a) 18 ° (b) 54 ° (c) 126° (d) 108 °**

**Solution – By angle sum property , we get **

**1x + 3x + 7x+ 9x = 180°**

**20x = 360°**

**x = 360/20**

**x = 18°**

**The angles of the**

**1x = 18°**

**3x = 54°**

**7x = 126°**

**9x = 162°**

** Answer — Hence the only incorrect option is 108° (option d)**

** **

**Q -2 Look at the adjoining figure of a ◻ABCD and name its following :**

**A. A pairs of adjacent sides**

# B. A pairs of opposite sides

C. A pairs of adjacent angles

D. A pairs opposite angles

**Answer :– (A) A Pair of adjacent angles are :-**

**AB , AD**

**BA , BC**

**CB , CD**

**DA , DC**

**(B) — A Pair of opposite sides —**

**AB , CD**

**BC , AD**

**(C) — A Pair of adjacent angles —**

**∠ A , ∠ B**

**∠ B ∠ C**

**∠ C ∠ D**

**∠ D ∠ A**

**(D) — A Pair of opposite angles**

**∠ A ∠ C**

**∠ B ∠ D**

**Q -3 In the figure of Q. no 2, the two diagonals of the quadrilateral ABCD.**

**Answer — Two diagonals of quadrilateral ABCD are: —**

**AC and BD**

**Q -4 Join Q and S in the adjoining figure and prove that :–**

**∠ P+ ∠ Q +∠ R +∠ S = 360°**

**Solution — In △ PQR**

**∠ 1 +**

**∠ 2 +**

**∠ 6 = 180**

**°**

**—— (1)**

**In**

**△ QRS**

**∠ 3 +**

**∠ 4 +**

**∠ 5 = 180**

**°**

**————(2)**

**Adding equation (1) and (2) we get**

**∠ 1 + (**

**∠ 2 +**

**∠ 3) +**

**∠ 4 + (**

**∠ 5 +**

**∠ 6) = 180**

**°**

**+180**

**°**

**∠ P +**

**∠ Q +**

**∠ R +**

**∠ S = 360**

**°**

**Hence proved .**

**Q –5 The Three angles of a quadrilateral are equal . If the measure of the fourth angle is 120°, what is the measure of the equal angles .**

**Solution — Let the equal angles be x**

**fourth angle = 120**

**°**

**by angle su**

**m property , we have**

**x + x + x + 120° = 360**

**°**

**3x + 120**

**°**

**= 360**

**3x = 360**

**°**

**– 120**

**°**

**3x = 240**

**°**

**x = 240 / 3**

**x = 80**

**°**

**Hence each angle are 80**

**°**

**each and fourth angle is 120**

**°**

**Q — 6 Two angles of a quadrilateral are of measure 75**

**°**

**each and the other two angle are equal. What is the measure of either of these two equal angles ?**

**Answer:–**

**Q — 7 If three angles of a quadrilateral are 20**

**°**

**, 90**

**°**

**and 90**

**°**

**, find the fourth angle of the quadrilateral .**

**Answer –7**

**Question — 8 The measure of two adjacent angles of a quadrilateral are 85 and 115 and the other two angles are equal . Find the measure of each angles.**

**Answer — **

Q –9 Four angles of a quadrilateral are 2: 3: 4: 1. Find the angles

**Answer — Given ratio = 2 : 3 : 4 : 1**

**let the measure of four angles = 2x , 3x , 4x , x by angle sum property**

**2x + 3x + 4x + x = 360°**

**10x = 360**

**°**

**x = 360 / 10**

**x = 36**

**°**

**2x = 2 ✖ 36 = 72**

**°**

**3x = 3**

**✖ 36 = 108**

**°**

**4x = 4**

**✖ 36 = 144**

**°**

**x = 36**

**°**

**Q — 10 The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. What is the measure of the four angles separately ?**

**Answer — Given ratio = 1: 2 : 3 : 4**

**let the measure of four angles = x , 2x , 3x , 4x by angle sum property**

**x + 2x + 3x + 4x = 360**

**°**

**10x = 360**

**°**

**x = 360 / 10**

**x = 36**

**°**

**2x = 2 ✖ 36 = 72**

**°**

**3x = 3**

**✖ 36 = 108**

**°**

**4x = 4**

**✖ 36 = 144**

**°**

**Q — 11 The four angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the angles of the quadrilateral.**

**Q –12 In the adjoining figure , E is a point in the interior of ∠ AOB , such that EC ⟂ OB and ED ⟂ OA. If ∠ AOB = 36° ,what is the measure of ∠ CED ?**

**Q — 13 The sum of two angles of a quadrilateral is 150° and the other angles are in the ratio 2 : 3. Find the measure of each angle . **

### ** ASSIGNMENT – 7.2**

Q – 1 Multiple Choice Questions (MCQ)

##### (i) Two adjacent sides of a parallelogram are 3cm and 4cm respectively. The perimeter of the parallelogram is

(a) 12cm (b) 13cm (c) 14cm (d) 15cm

Solution :–

Length of two adjacent sides = 4 cm and 3 cm i.e.,

l=4 cm and b = 3 cm

∴ Perimeter = 2(l+b)

= 2 ( 4 + 3 ) c m

= 2 × 7 = 14 c m

###### (ii) The long side of a parallelogram is 8cm. If the shorter side is 3/4 of the longer side, the perimeter of the parallelogram is

(a) 28cm (b) 26cm (c) 24cm (d) 22cm

Solution :- Length is 8cm

Breadth is 3/4 × 8 = 6cm

Therefore perimeter= 2(l+b) = 2(8+6)

= 21 × 4 = 28cm

##### (iii) In a parallelogram ABCD , ∠A = 45 ° , then the other angles are

(a) 145 ° , 45 ° , 145 °

##### (b) 150 ° , 40 ° , 150 °

##### (c) 135 ° , 45 ° ,135 °

##### (d) None of these

Solution:–

###### For a parallelogram ABCD,

∠A=∠C , ∠B=∠D

by angle sum property ∠A+∠B+∠C+∠D=360°

∠A= 45 ° (given)

∠C = 45 °

45 ° + ∠B + 45 ° + ∠D = 360°

90° + 2 ∠B = 360°

∠B=∠D=135°

##### Q –2 The longer side of a parallelogram is 8.4 cm and the shorter side is half of the longer side . Find the perimeter of the parallelogram.

###### Solution:–

###### Longer side = 8.4cm

###### Shorter side = 8.4/2 = 4.2 cm

###### Perimeter=2( l+b)

###### = 2 (8.4+4.2) = 25.2cm

###### Q–3 The ratio the adjacent sides of a parallelogram is as 2 : 3, and its perimeter is 40cm . Find the sides of the parallelogram.

##### Solution :–

Let adjacent sides of the parallelogram be 2x and 3x

##### Then , 2x + 3x + 2x + 3x = 10x

##### perimeter is given = 40cm

##### 10x = 40cm

##### x = 40/10 = 4cm

##### 2x = 2 × 4 = 8cm

##### 3x = 3 × 4 = 12cm

##### Q–4 The ratio the adjacent sides of a parallelogram is as 2 : 3, and its perimeter is 60cm . Find the sides of the parallelogram.

##### Solution:–

###### Ratio = 2:3

Put x in the ratio,

Length (l) = 2x

Breadth (b) = 3x

Perimeter = 60 cm

Perimeter = 2( l + b )

2( 2x + 3x ) = 60 cm

2 × 5x = 6ocm

10x = 60cm

x = 60 /10

x = 6cm

The value of x is 6 cm

Length = 2x = 2×6 = 12 cm

Breadth = 3x = 3×6 = 18 cm

##### Q–5 The perimeter of a parallelogram is 150cm. One of its sides is greater than the other by 33cm. Find the lengths of the sides of the parallelogram.

##### Solution :–

let the breadth be = x cm

and length be = x +33 cm

Given perimeter = 150 cm

Perimeter = 2 ( l + b )

2 ( x + x + 33 ) = 150 cm

2 ( 2x + 33 ) = 150 cm

4x + 66 = 150 cm

4x = 150 – 66

4x = 84 cm

x = 84 / 4

x = 21 cm

breadth = 21 cm

length = 21 + 33 = 54 cm

##### Q–6 Two adjacent angles of a parallelogram are in the ratio 4:5 . Find the measure of all the angles.

Solution:–

The ratio between two adjacent angles of a || gm ABCD are in the ratio 4 : 5

Let ∠ A = 4 x

and ∠ B = 5 x

But ∠ A + ∠ B = 180°

⇒ 4 x + 5 x = 180°

⇒ 9 x = 180°

∴ x = 180 / 9 = 20°

##### ∴ ∠ A = 4 x = 4 × 20° = 80 °

∠ B = 5 x = 5 × 20° = 100°

∠C = 4x = 4 × 20° = 80°

∠D = 5 x = 5 × 20° = 100°

##### Q–7 Two adjacent angles of a parallelogram are in the ration of 2:1. Find the measure of each angle.

Solution:–

##### Q –8 Two adjacent angles of a parallelogram are in the ration of 7:2. Find the measure of all the angles of parallelogram.

Solution:–

##### Q–9 In the adjoining Figure , ABCD is a parallelogram if ∠DAB = 85° and ∠DBA = 60°, then calculate:–

(i) ∠CDB (ii) ∠ABD

Solution–

###### (i) In parallelogram ABCD, ∠A = ∠C = 85°

. In ∆BCD,

∠DBC + ∠BCD + ∠CDB = 180°

i.e. 60° + 85° + ∠CDB = 180°

###### or 145° + ∠CDB = 180°

or ∠CDB = 180° – 145° = 35°

###### (ii) ABCD is a parallelogram,

so AB || DC, BD is a transversal.

∴ ∠ABD = ∠CDB [Alternate interior opp. angles]

i.e. ∠ABD = ∠CDB = 35°

**Class – 8 Maths (Prachi Publication) chapter – 7**

#### ** ASSIGNMENT – 7.3 **

### Q – 1 Multiple choice Questions (MCQ) Choose the correct option .

### (i) Which of the following is not correct for a rhombus ?

(a) Diagonals bisect each other. (b) All angles are 90°

(c) Diagonals perpendicular to each other (d) All sides equal.

Solution :– option b , All angles are 90°

(ii) If one of the angles formed by diagonals and adjacent sides of a rhombus is 20° , all four angles of the rhombus are

(a) 40° , 140° , 40° , 140° (b) 50° , 130° , 50° , 130°

(c) 60° , 120° , 60° , 120° (d) None of these.

Solution :–

** ****Option A , (a) 40° , 140° , 40° , 140° **

** **

**iii) If a perpendicular drawn from an obtuse – angled vertex of a rhombus to the opposite side bisects the side , all angles of the rhombus are :–**

**(a) 70° , 110° , 70° , 110° (b) 40° , 140° , 40° , 140° **

**(c) 50° , 130° , 50° , 130° (d) 60° , 120° , 60° ,120°**

**Solution :– **

** **

**Q– 2 For each of the following statements , state whether the statement is true (T) or false (F) :- **** Solution:–**

**(i) Every square is a rectangle. — True**

**(ii) Every parallelogram is a rhombus. —- False**

**(iii) Every rectangle is a parallelogram. —— True**

**(iv) Every rectangle is a square. ———– False**

**(v) Every square is a rhombus,———- True**

**(vi) Every rhombus is a square.————-False**

**(vii) Every square is a parallelogram. ——-True**

**(viii) Every rhombus is a parallelogram. —–True**

**(ix) Every parallelogram is a rectangle.———-False**

**(x) Every parallelogram is a trapezium.———-True**

**(xi) Every parallelogram is a square.————-False**

**(xii) Every square is a trapezium. —————-True**

**(xiii) Every trapezium is a square. —————–False**

**(xiv) Every trapezium is a parallelogram.———False**

**Q–3 Which of the following statements are true (T) or false (F) for a rectangle and a square:–**

**(ii) It has two pairs of opposite sides of equal length.**

**(iii) Its diagonals bisect each other.**

**(iv) Its diagonals are perpendicular and bisect each other.**

**(v) Its diagonals are equal and perpendicular to each other.**

**(vi) Its diagonals are perpendicular to each other.**

**(vii)) Its diagonals are equal , perpendicular and bisect each other.**

**(viii) All of its angles are equal.**

**(ix) Its diagonals are equal and bisect each other.**

**(x) Its diagonals are equal.**

**Solution:–**

**Rectangle**

**(i) F**

**(ii) T**

**(iii) T**

**(iv) F**

**(v) F**

**(vi) F**

**(vii) F**

**(viii) T**

**(ix) T**

**(x) T**

(i) T

(ii) T

(iii) T

(iv) T

(v) T

(vi) T

(vii) T

(viii) T

(ix) T

**Q – 4 Which of the following statements are (T) or (F) for a rhombus ?**

**Answer:–**

(ii) It has two pairs of parallel sides………..T

**Q–5 ABCD is parallelogram. What special name will you give it, if the following additional facts are known ?**

**(i) AB = AD (ii) ∠DAB = 90° (iii) AB = AD and ∠DAB = 90°**

**Q–6 In the adjoining figure , ABCD is a rhombus. **

**find the measure of the following angles , if ∠ACB = 30**

**°**

**(i) ∠BOC (ii) ∠CBO**

**( iii) ∠OAD (iv) ∠ABO**

**Solution :–**

**Q — 7 In a given rectangle ABCD , diagonals AC and BD intersect at O . If ∠COD = 120° find ∠ OBA .**

**Solution :–**

**Q – 8 In the given figure , prove that the diagonals of a rectangle are equal .**

**Solution :-**–

**Q — 9 Prove that diagonals of a rhombus bisect each other at right angles as given in the adjoining figure .**

**Solution:–**

**Q– 10 Prove that a rhombus with one angle 90° is a square.**

**Q– 11 Show that the four triangles as shown in the adjoining figure formed by diagonals and sides of rhombus are congruent. **

**Solution:–**

**. Diagonals of a rhombus bisect each other at right angles.**

**In ∆s COD and BOC, we have**

**DC = BC [Sides of rhombus]**

**OC = OC [Common]**

**∴ ∆CDO ≅ ∆BOC [By SAS]**

**Similarly, we can prove that ∆BOC ≅ ∆AOB,**

**∆AOB ≅ ∆AOD,**

**∆AOD ≅ ∆COD.**

**Thus, ∆COD ≅ ∆BOC ≅ ∆AOB ≅ ∆AOD.**

**Hence, the four triangles are congruent. Hence proved.**

**Q–12 In the given figure , ABCD is a rectangle . BM and DN are perpendiculars to AC from B and D respectively. Prove that AN=CM.**

**Solution:—**