## NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

- Class 8 Maths Understanding Quadrilaterals Exercise 3.1
- Class 8 Maths Understanding Quadrilaterals Exercise 3.2
- Class 8 Maths Understanding Quadrilaterals Exercise 3.3
- Class 8 Maths Understanding Quadrilaterals Exercise 3.4
- Understanding Quadrilaterals Class 8 Extra Questions

**NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Exercise 3.1**

Question 1.

Given here are some figures.

Classify each of the above figure on the basis of the following:

(a) Simple curve

(b) Simple closed curve

(c) Polygon

(d) Convex polygon

(e) Concave polygon

Solution:

(a) Simple curve: (1), (2), (5), (6) and (7)

(b) Simple closed curve: (1), (2), (5), (6) and (7)

(c) Polygon: (1) and (2)

(d) Convex polygon: (2)

(e) Concave polygon: (1)

Question 2.

How many diagonals does each of the following have?

(a) A convex quadrilateral

(b) A regular hexagon

(c) A triangle

Solution:

(a) In Fig. (i) ABCD is a convex quadrilateral which has two diagonals AC and BD.

(b) In Fig. (ii) ABCDEF is a regular hexagon which has nine diagonals AE, AD, AC, BF, BE, BD, CF, CE and DF.

(c) In Fig. (iii) ABC is a triangle which has no diagonal.

Question 3.

What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and verify)

Solution:

In the given figure, we have a quadrilateral ABCD. Join AC diagonal which divides the quadrilateral into two triangles ABC and ADC.

In ∆ABC, ∠3 + ∠4 + ∠6 = 180°…(i) (angle sum property)

In ∆ADC, ∠1 + ∠2 + ∠5 = 180° …(ii) (angle sum property)

Adding, (i) and (ii)

∠1 + ∠3 + ∠2 + ∠4 + ∠5 + ∠6 = 180° + 180°

⇒ ∠A + ∠C + ∠D + ∠B = 360°

Hence, the sum of all the angles of a convex quadrilateral = 360°.

Let us draw a non-convex quadrilateral.

Yes, this property also holds true for a non-convex quadrilateral.

Question 4.

Examine the table. (Each figure is divided into triangles and the sum of the angles reduced from that).

What can you say about the angle sum of a convex polygon with number of sides?

(a) 7

(b) 8

(c) 10

(d) n

Solution:

From the above table, we conclude that the sum of all the angles of a polygon of side ‘n’

= (n – 2) × 180°

(a) Number of sides = 7

Angles sum = (7 – 2) × 180° = 5 × 180° = 900°

(b) Number of sides = 8

Angle sum = (8 – 2) × 180° = 6 × 180° = 1080°

(c) Number of sides = 10 Angle sum = (10 – 2) × 180° = 8 × 180° = 1440°

(d) Number of sides = n

Angle sum = (n – 2) × 180°

Question 5.

What is a regular polygon? State the name of a regular polygon of

(i) 3 sides

(ii) 4 sides

(iii) 6 sides

Solution:

A polygon with equal sides and equal angles is called a regular polygon.

(i) Equilateral triangle

(ii) Square

(iii) Regular Hexagon

Question 6.

Find the angle measure x in the following figures:

Solution:

(a) Angle sum of a quadrilateral = 360°

⇒ 50° + 130° + 120° + x = 360°

⇒ 300° + x = 360°

⇒ x = 360° – 300° = 60°

(b) Angle sum of a quadrilateral = 360°

⇒ x + 70° + 60° + 90° = 360° [∵ 180° – 90° = 90°]

⇒ x + 220° = 360°

⇒ x = 360° – 220° = 140°

(c) Angle sum of a pentagon = 540°

⇒ 30° + x + 110° + 120° + x = 540° [∵ 180° – 70° = 110°; 180° – 60° = 120°]

⇒ 2x + 260° = 540°

⇒ 2x = 540° – 260°

⇒ 2x = 280°

⇒ x = 140°

(d) Angle sum of a regular pentagon = 540°

⇒ x + x + x + x + x = 540° [All angles of a regular pentagon are equal]

⇒ 5x = 540°

⇒ x = 108°

Question 7.

(a) Find x + y + z

(b) Find x + y + z + w

Solution:

(a) ∠a + 30° + 90° = 180° [Angle sum property]

⇒ ∠a + 120° = 180°

⇒ ∠a = 180° – 120° = 60°

Now, y = 180° – a (Linear pair)

⇒ y = 180° – 60°

⇒ y = 120°

and, z + 30° = 180° [Linear pair]

⇒ z = 180° – 30° = 150°

also, x + 90° = 180° [Linear pair]

⇒ x = 180° – 90° = 90°

Thus x + y + z = 90° + 120° + 150° = 360°

(b) ∠r + 120° + 80° + 60° = 360° [Angle sum property of a quadrilateral]

∠r + 260° = 360°

∠r = 360° – 260° = 100°

Now x + 120° = 180° (Linear pair)

x = 180° – 120° = 60°

y + 80° = 180° (Linear pair)

⇒ y = 180° – 80° = 100°

z + 60° = 180° (Linear pair)

⇒ z = 180° – 60° = 120°

w = 180° – ∠r = 180° – 100° = 80° (Linear pair)

x + y + z + w = 60° + 100° + 120° + 80° = 360°.

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