NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.1

These NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals Ex 3.1 Questions and Answers are prepared by our highly skilled subject experts.

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

Question 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 are: (1), (2), (5) (6) and (7)
(b) Simple closed curves are: (1), (2), (5), (6) and (7) and (4)
(c) Polygons are: (1) and (2)
(d) Convex polygon is (2)
(e) Concave polygon is (1) and (4)

Question 2.
How many diagonals does each of the following have?
(b) A regular hexagon
(c) A triangle
Solution:
Note: Number of diagonals in a polygon having n sides
n-side = $$\left[\frac{n(n-1)}{2}-n\right]$$
(a) In quadrilateral, number of sides (n) = 4
Number of diagonals = 2
$$\left[\frac{4(4-1)}{2}-4=\frac{4 \times 3}{2}-4=6-4=2\right]$$

(b) In a regular hexagon number of sides, (n) = 6
$$\left[\frac{6(6-1)}{2}-6=\frac{6 \times 5}{2}-6=15-6=9\right]$$

(c) In a triangle, number of sides (n) = 3
∴ Number of diagonals = $$\frac{n(n-1)}{2}-n$$
= $$\frac{3(3-1)}{2}$$ – 3
= $$\frac{3 \times 2}{2}$$ – 3
= 3 – 3
= 0

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 try!)
Solution:
The sum of the measures of the angles of a convex quadrilateral is 360°
Yes, this property holds, even if the quad-rilateral is not convex.

Question 4.
Examine the table.
Each (figure is divided into triangles and the sum of the 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) When n = 7
Sum of interior angles of a polygon = (7 – 2) × 180°
= 5 × 180°
= 900°

(b) When n = 8
Sum of interior angles of a polygon = (8 – 2) × 180°
= 6 × 180°
= 1080°

(c) When n = 10
Sum of interior angles of a polygon having 10 sides = (10 – 2) × 180°
= 8 × 180°
= 1440°

(d) When n = n
Sum of interior angles of a polygon = (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 is said to be a regular polygon if
(a) The measures of its interior angles are equal
(b) The length of its sides are equal
The name of a regular polygon having
(i) 3 sides is ‘equilateral triangle’
(ii) 4 sides is ‘square’.
(iii) 6 sides is ‘regular hexagon’.

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

Solution:
(a) The sum of interior angles of a quadrilateral is 360°
∴ x + 120° + 130° + 50° = 360°
⇒ x + 300° = 360°
⇒ x = 360° – 300°
⇒ x = 60°

(b) The sum of the interior angles of a quadrilateral is 360°
∴ x + 70° + 60° + 90° = 360°
⇒ x + 220° = 360°
⇒ x = 360° – 220°
⇒ x = 140°

(c) Interior angles are 30°, x°, (180 – 60°), (180° – 70°) and x°
i.e., 30°, x°, 120°, 110° and x°
The given figure is a pentagon.
Sum of interior angles of a pentagon = 540°
∴ 30° + x ° + 120° + 110° + x = 540°
⇒ 2x° + 260° = 540°
⇒ 2x° = 540° – 260°
⇒ 2x°= 280°
⇒ x = $$\frac{280^{\circ}}{2}$$
⇒ x = 140°
The measure of x is 140°.

(d) It is a regular pentagon.
Sum of all the interior angles of regular pentagon = 540°
It’s each angle is equal to x°.
x° + x° + x° + x° + x° = 540°
⇒ 5x° = 540°
⇒ x = $$\frac{540^{\circ}}{5}$$
⇒ x = 108°
The measure of x is 108°.

Question 7.

(a) Find x + y + z

(b) Find x + y + z + w
Solution:
(a) x + 90° = 180° (Linear Pair)
x = 180° – 90° = 90°
y = 30° + 90° [∵ exterior angle of a triangle is equal to the sum of interior opposite angles]
⇒ y = 120°
z + 30° = 180° (Linear pair)
⇒ z = 180° – 30° = 150°
∴ x + y + z = 90° + 120° + 150° = 360°

(b) The sum of interior angles of a quadrilateral = 360°
⇒ ∠1 + 120° + 80° + 60° = 360°
⇒ ∠1 + 260° = 360°
⇒ ∠1 = 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°
⇒ z = 120°
w + ∠1 = 180° (Linear Pair)
⇒ w + 100°= 180°
⇒ w = 180° – 100° = 80°
Thus x + y + z + w = 60° + 100° + 120° + 80° = 360°

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