Class 8th maths Chapter 3 Understanding of Quadrilaterals | 8th class mathematics

NCERT SOLUTIONS FOR CLASS 8 MATHS CHAPTER 3
Understanding of Quadrilaterals
8th class mathematics Ex 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

Solution:
(a) Simple curve – 1, 2, 5, 7
(b) Simple closed curve – 1, 2, 5, 6, 7
(c) Polygon – 1, 2
(d) Convex polygon – 2 
(e) Concave polygon – 1, 4

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

(a) A convex quadrilateral

(b) A regular hexagon

(c) A triangle

Solution:
(a) A convex quadrilateral – 2
(b) A regular hexagon – 9
(c) A triangle – 0

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!)
Solution: The sum of the measures of the angles of a convex quadrilateral is 360°.

The above quadrilateral is not convex. This quadrilateral has been divided into two triangles. The value of each triangle will be 180°. Therefore, the sum of the angles of this quadrilateral will also be 360°.

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

Figure
Side	3	4	5	6
Angle sum	180º	2 × 180°= (4 – 2) × 180°	3 × 180°= (5 – 2) × 180°	4 × 180°= (6 – 2) × 180°

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) Number of sides (x) = 7
The sum of all the interior angles of a polygon can be found by the following formula = (x – 2) × 180°
Put the value of the x
= (7 – 2) × 180
= 5 × 180
= 900°

(b) Number of sides (x) = 8
Sum of all interior angles of polygon = (x – 2) × 180°
Put the value of the x
= (8 – 2) × 180
= 6 × 180
= 1080°

(c) Number of sides (x) = 10
Sum of all interior angles of polygon = (x – 2) × 180°
Put the value of the x
= (10 – 2) × 180
= 8 × 180
= 1440°

(d) Number of sides (x) = n
Sum of all interior angles of polygon = (x – 2) × 180°
Put the value of the x
= (n – 2) × 180

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

3 sides

4 sides

6 sides

Solution: 
Regular Polygon : A polygon whose all sides and interior angles are equal is called a regular polygon.
3 Sides : A polygon whose all three sides are equal is called a triangle (equilateral triangle).
4 Sides : A polygon whose all four sides are equal is called a quadrilateral (square).
6 Sides : A polygon whose all six sides are equal is called a regular hexagon.

6. Find the angle measure x in the following figures.
(a)
Solution: 
∵ Sum of four angles of quadrilateral = 360°
Hence 50° + 130° + 120° + x = 360°
300° + x = 360°
x = 360° – 300°
x = 60°

(b)
Solution:
y + z = 180° (linear angle pair)
90° + z = 180°
z = 180° – 90°
z = 90°
∵ Sum of four angles of quadrilateral = 360° 
70° + 60° + 90° + x = 360°
220° + x = 360°
x = 360° – 220°
x = 140°

(c)

Solution:
70° + y = 180° (linear angle pair)
y = 180° – 70°
y = 110°
60° + z = 180° (linear angle pair)
z = 180° – 60°
z = 120°
∵ sum of all interior angles of a pentagon = (n − 2) × 180°
n = 5
= (5 – 2) × 180°
= 3 × 180°
= 540°
therefore, 110° + 120° + x + x + 30° = 540°
260° + 2x = 540°
2x = 540° – 260°
2x = 280°
x = 280° ÷ 2
x = 140°

(d)
Solution: 
∵ sum of the five angles of a pentagon = 540° 
therefore, x + x + x + x + x = 540°
5x = 540°
x = 540° ÷ 5
x = 108°

7. (a)
Solution: 
z + 30° = 180° (linear angle pair)
z = 180° – 30°
z = 150°
x + 90° = 180° (linear angle pair)
x = 180° – 90°
x = 90°
The sum of the three exterior angles of a triangle = 360°

(b) Find x + y + z + w

x + 120° = 180° (linear angle pair)
x = 180° – 120°
x = 60°
y + 80° = 180° (linear angle pair)
y = 180° – 80°
y = 100°
z + 60° = 180° (linear angle pair)
z = 180° – 60°
z = 120°
∵ Sum of four angles of quadrilateral = 360°
x + y + z + w = 360°
60° + 100° + 120° + w = 360°
280° + w = 360°
w = 360° – 280°
w = 80°
So, x + y + z + w = 360°

EXERCISE 3.2

1. Find x in the following figures.
(a)
Solution:
∵ Sum of exterior angles of any polygon = 360°
therefore, 125° + 125° + x = 360°
250° + x = 360°
x = 360° – 250°
x = 110°

(b) Solution:
∵ Sum of exterior angles of pentagon = 360°
Therefore, 70° + x + 90° + 60° + 90° = 360°
310° + x= 360°
x = 360° – 310°
x = 50°

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

(i) 9 sides

(ii) 15 sides

Solution:
(i) 9 sides
measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{n}\) n = 9
Therefore, the measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{9}\) = 40°

(ii) 15 sides
measure of each exterior angle of a regular polygon = = \(\displaystyle \frac{{360}}{n}\) n = 15 
Therefore, the measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{15}\) = 24°