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

**NCERT Solutions for Class 8 Maths Chapter 3 Exercise 3.1 – Understanding Quadrilaterals, has been designed by the NCERT to test the knowledge of the student on the following topics : **

- Introduction
- Polygons

Classification of polygons

Diagonals

Convex and concave polygons

Regular and irregular polygons

Angle sum property

**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**

**Solution:**

(a) Simple curves: 1, 2, 5, 6, 7

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

(c) Polygons: 1, 2

(d) Convex polygon: 2

(e) Concave polygon: 1

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

**(a) A convex quadrilateral**

**(b) A regular hexagon**

**(c) A triangle**

**Solution:**

(a) A convex quadrilateral has 2 diagonals,

(b) A regular hexagon has 9 diagonals.

(c) A triangle has no diagonals.

**3. What is the sum of the measures of angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex?**

**Solution:**

A convex quadrilateral can be divided into two triangle by one of the diagonals

∴ Angle sum of convex quadrilateral

= Angle sum of two triangles

= 2 × 180°

= 360°

Similarly, a non-convex quadrilateral can also be divided into two triangles

Hence, the property holds true for non-convex quadrilaterals also.

**4. Examine the table. (Each figure is divided into triangles and the sum of angles deduced 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:**

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

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

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

(d) Angle sum = (n – 2) × 180°

**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) A regular polygon of 3 sides is called an **equilateral triangle**.

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

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

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

**a)**

** **

**Solution:**

Using angle sum property of quadrilateral

50° + 130° + 120° + x = 360°

300° + x = 360°

x = 60°

**b) **

**Solution:**

Using angle sum property of quadrilateral

70° + 60° + 90° + x = 360°

220° + x = 360°

x = 360° – 220°

x = 140°

**c)**

** **

**Solution:**

The interior angles of the base of the pentagon will be 110°(= 180° – 70°) and 120°(= 180° – 60°)

Angle sum of pentagon = (5 – 2) × 180° = 3 × 180° = 540°

30° + x + 110° + 120° + x = 540°

260° + 2x = 540°

2x = 540° – 260°

2x = 280°

x = 140°

**d) **

**Solution:**

Angle sum of pentagon = (5 – 2) × 180° = 3 × 180° = 540°

x + x + x + x + x = 540°

5x = 540°

x = 108°

**7. (a) Find x + y + z**

**Solution:**

Exterior angle = 180° – Interior angle

∴ z = 180° – 30° = 150°

x = 180° – 90° = 90°

Interior angle = 180° – Exterior angle

∴ Unknown interior angle of triangle = 180° – y

Using angle sum property of triangle

90° + 30° + (180° – y) = 180°

y = 90° + 30° + 180° – 180°

y = 90° + 30°

y = 120°

Therefore, x + y + z = 90° + 120° + 150° = 360°

**(b) Find x + y + z + w**

**Solution:**

Exterior angle = 180° – Interior angle

∴ x = 180° – 120° = 60°

y = 180° – 80° = 100°

z = 180° – 60° = 120°

Interior angle = 180° – Exterior angle

∴ Unknown interior angle of quadrilateral = 180° – w

Angle sum of quadrilateral = (4 – 2) × 180° = 2 × 180° = 360°

60° + 80° + 120° + (180° – w) = 360°

w = 60° + 80° + 120° + 180° – 360°

w = 440° – 360°

w = 80°

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

**The next Exercise for** **NCERT Solutions for Class 8 Maths Chapter 3 Exercise 3.2 – Understanding Quadrilaterals can be accessed by clicking here **

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

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