#### NCERT Solutions for Class 8th: Ch 3 Understanding Quadrilaterals Maths Part-I

Page No: 41**Exercise 3.1**

1. Given here are some figures.

Classify each of them on the basis of the following.

(a) Simple curve (b) Simple closed curve (c) Polygon(d) Convex polygon (e) Concave polygon

**Answer**

(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

2. How many diagonals does each of the following have?

(a) A convex quadrilateral (b) A regular hexagon (c) A triangle

**Answer**

(a) A convex quadrilateral: It has 2 diagonals.

(b) A regular hexagon

(c) A triangle

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 try!)

**Answer**

Let ABCD be a convex quadrilateral. We observe that the quadrilateral ABCD formed by two triangles i.e. ΔADC and ΔABC.

Since, we know that sum of interior angles of triangle is 180°. Thus, the sum of the measures of the angles is 180° + 180° = 360°

Let us take another quadrilateral ABCD which is not convex and join BC which divides it into two triangles ΔABC and ΔBCD.

In ΔABC,

∠1 + ∠2 + ∠3 = 180° (angle sum property of triangle)

In ΔBCD,

∠4 + ∠5 + ∠6 = 180° (angle sum property of triangle)

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

⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°

⇒ ∠A + ∠B + ∠C + ∠D = 360°Thus, this property hold if the quadrilateral is not convex.

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

(a) 7 (b) 8 (c) 10 (d) n

**Answer**

The angle sum of a polygon having side n = (n-2)×180°

(a) 7

Here, n = 7

Thus, angle sum = (7-2)×180° = 5×180° = 900°

(b) 8

Here, n = 8

Thus, angle sum = (8-2)×180° = 6×180° = 1080°

(c) 10

Here, n = 10

Thus, angle sum = (10-2)×180° = 8×180° = 1440°

(d) n

Here, n = n

Thus, angle sum = (n-2)×180°

Page No: 42

5. What is a regular polygon?

State the name of a regular polygon of

(i) 3 sides (ii) 4 sides (iii) 6 sides

**Answer**

A polygon having sides of equal length and angles of equal measures is called regular polygon.

(i) A regular polygon of 3 sides is equilateral triangle.

(ii) A regular polygon of 4 sides is square.

(iii) A regular polygon of 6 sides is regular hexagon.

6. Find the angle measure x in the following figures.

**Answer**

(a) The figure is having 4 sides. Hence, it is a quadrilateral.

Sum of angles of the quadrilateral = 360°

⇒ 50° + 130° + 120° + x = 360°

⇒ 300° + x = 360°

⇒ x = 360° - 300° = 60°

(b) The figure is having 4 sides. Hence, it is a quadrilateral. Also, one side is perpendicular forming right angle.

Sum of angles of the quadrilateral = 360°

⇒ 90° + 70° + 60° + x = 360°

⇒ 220° + x = 360°

⇒ x = 360° - 220° = 140°

(c) The figure is having 5 sides. Hence, it is a pentagon.

Two angles at the bottom are linear pair.

Therefore, 180° - 70° = 110°

180° - 60° = 120°

⇒ 30° + 110° + 120° + x + x = 540°

⇒ 260° + 2x = 540°

⇒ 2x = 540° - 260° = 280°

⇒ x = 280°/2 = 140°

(d) The figure is having 5 equal sides. Hence, it is a regular pentagon. Thus, its all angles are equal.

5x = 540°

⇒ x = 540°/5

⇒ x = 108°

7.

**Answer**

(a) Sum of all angles of triangle = 180°

One side of triangle = 180°- (90°

x + 90° = 180° ⇒ x = 180° - 90° = 90°

y + 60° = 180° ⇒ y = 180° - 60° = 120°

z + 30° = 180° ⇒ z = 180° - 30° = 150°

x + y + z = 90°

(b) Sum of all angles of quadrilateral = 360°

One side of quadrilateral = 360°- (60° + 80° + 120°) = 360° - 260° = 100°

x + 120° = 180° ⇒ x = 180° - 120° = 60°

y + 80° = 180° ⇒ y = 180° - 80° = 100°

z + 60° = 180° ⇒ z = 180° - 60° = 120°

w + 100° = 180° ⇒ w = 180° - 100° = 80°

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

Page No: 44

**Exercise 3.2**

1. Find x in the following figures.

**Answer**

(a)

125° + m = 180° ⇒ m = 180° - 125° = 55° (Linear pair)

125° + n = 180° ⇒ n = 180° - 125° = 55° (Linear pair)x = m + n (exterior angle of a triangle is equal to the sum of two opposite interior two angles)

⇒ x = 55°

(b)

Two interior angles are right angles = 90°

70° + m = 180° ⇒ m = 180° - 70° = 110° (Linear pair)

60° + m = 180° ⇒ m = 180° - 60° = 120° (Linear pair) The figure is having five sides and is a pentagon.

Thus, sum of the angles of pentagon = 540°

90° + 90° + 110° + 120° + y = 540°

⇒ 410° + y = 540° ⇒ y = 540° - 410° = 130°

x + y = 180° (Linear pair)

⇒ x + 130° = 180°

⇒ x = 180° - 130° = 50°

2. Find the measure of each exterior angle of a regular polygon of

(i) 9 sides (ii) 15 sides

**Answer**

Sum of angles a regular polygon having side n = (n-2)×180°

(i) Sum of angles a regular polygon having side 9 = (9-2)×180°

= 7×180° = 1260°

Each interior angle = 1260°/9 = 140°

Each exterior angle = 180° - 140° = 40°

Or,

Each exterior angle = Sum of exterior angles/Number of sides = 360°/9 = 40°

(i) Sum of angles a regular polygon having side 15 = (15-2)×180°

= 13×180° = 2340°

Each interior angle = 2340°/15 = 156°

Each exterior angle = 180° - 156° = 24°

Or,

Each exterior angle = Sum of exterior angles/Number of sides = 360°/15 = 24°

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

**Answer**

Each exterior angle = Sum of exterior angles/Number of sides

24° = 360°/Number of sides

⇒ Number of sides = 360°/24° = 15

Thus, the regular polygon have 15 sides.

4. How many sides does a regular polygon have if each of its interior angles is 165°?

**Answer**

Interior angle = 165°

Exterior angle = 180° - 165° = 15°

Number of sides = Sum of exterior angles/exterior angle

⇒ Number of sides = 360°/15° = 24

Thus, the regular polygon have 24 sides.

5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

(b) Can it be an interior angle of a regular polygon? Why?

**Answer**

(a) Exterior angle = 22°

Number of sides = Sum of exterior angles/exterior angle

⇒ Number of sides = 360°/22° = 16.36

No, we can't have a regular polygon with each exterior angle as 22° as it is not divisor of 360.

(b) Interior angle = 22°

Exterior angle = 180° - 22°= 158°

No, we can't have a regular polygon with each exterior angle as 158° as it is not divisor of 360.

6. (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

**Answer**

(a) Equilateral triangle is regular polygon with 3 sides has the least possible minimum interior angle because the regular with minimum sides can be constructed with 3 sides at least..

Since, sum of interior angles of a triangle = 180°

Each interior angle = 180°/3 = 60°

(b) Equilateral triangle is regular polygon with 3 sides has the maximum exterior angle because the regular polygon with least number of sides have the maximum exterior angle possible.

Maximum exterior possible = 180 - 60° = 120°

Go to Part-II (Exercise 3.3 and 3.4)

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