Rs Aggarwal 2018 for Class 8 Math Chapter 14 - Polygons

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Rs Aggarwal 2018 Solutions for Class 8 Math Chapter 14 Polygons are provided here with simple step-by-step explanations. These solutions for Polygons are extremely popular among Class 8 students for Math Polygons Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2018 Book of Class 8 Math Chapter 14 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2018 Solutions. All Rs Aggarwal 2018 Solutions for class Class 8 Math are prepared by experts and are 100% accurate.

Page No 182:

Question 1:

Find the measure of each exterior angle of a regular
(i) pentagon
(ii) hexagon
(iii) heptagon
(iv) decagon
(v) polygon of 15 sides.

Answer:

Exterior angle of an n-sided polygon = ${(\frac{360}{n})}^{o}$
(i) For a pentagon: $n = 5 ∴ (\frac{360}{n}) = (\frac{360}{5}) = 72^{o}$

(ii) For a hexagon: $n = 6 ∴ (\frac{360}{n}) = (\frac{360}{6}) = 60^{o}$

(iii) For a heptagon: $n = 7 ∴ (\frac{360}{n}) = (\frac{360}{7}) = 51 . 43^{o}$

(iv) For a decagon: $n = 10 ∴ (\frac{360}{n}) = (\frac{360}{10}) = 36^{o}$

(v) For a polygon of 15 sides: $n = 15 ∴ (\frac{360}{n}) = (\frac{360}{15}) = 24^{o}$

Page No 182:

Question 2:

Is it possible to have a regular polygon each of whose exterior angles is 50°?

Answer:

Each exterior angle of an n-sided polygon = ${(\frac{360}{n})}^{o}$
If the exterior angle is 50°, then:

$\frac{360}{n} = 50 \Rightarrow n = 7.2$

Since n is not an integer, we cannot have a polygon with each exterior angle equal to 50°.

Page No 182:

Question 3:

Find the measure of each interior angle of a regular polygon having
(i) 10 sides
(ii) 15 sides.

Answer:

For a regular polygon with n sides:
$Each interior angle = 180 - \{Each exterior angle\} = 180 - (\frac{360}{n})$

(i) For a polygon with 10 sides:
$Each exterior angle = \frac{360}{10} = 36^{o} \Rightarrow Each interior angle = 180 - 36 = 144^{o}$

(ii) For a polygon with 15 sides:
$Each exterior angle = \frac{360}{15} = 24^{o} \Rightarrow Each interior angle = 180 - 24 = 156^{o}$

Page No 182:

Question 4:

Is it possible to have a regular polygon each of whose interior angles is 100°?

Answer:

Each interior angle of a regular polygon having n sides = $180 - (\frac{360}{n}) = \frac{180 n - 360}{n}$

If each interior angle of the polygon is 100°, then:

$100 = \frac{180 n - 360}{n} \Rightarrow 100 n = 180 n - 360 \Rightarrow 180 n - 100 n = 360 \Rightarrow 80 n = 360 \Rightarrow n = \frac{360}{80} = 4.5$

Since n is not an integer, it is not possible to have a regular polygon with each interior angle equal to 100°.

Page No 182:

Question 5:

What is the sum of all interior angles of a regular
(i) pentagon
(ii) hexagon
(iii) nonagon
(iv) polygon of 12 sides?

Answer:

Sum of the interior angles of an n-sided polygon = $(n - 2) \times 180 °$

(i) For a pentagon:
$n = 5 ∴ (n - 2) \times 180 ° = (5 - 2) \times 180 ° = 3 \times 180 ° = 540 °$

(ii) For a hexagon:
$n = 6 ∴ (n - 2) \times 180 ° = (6 - 2) \times 180 ° = 4 \times 180 ° = 720 °$

(iii) For a nonagon:
$n = 9 ∴ (n - 2) \times 180 ° = (9 - 2) \times 180 ° = 7 \times 180 ° = 1260 °$

(iv) For a polygon of 12 sides:
$n = 12 ∴ (n - 2) \times 180 ° = (12 - 2) \times 180 ° = 10 \times 180 ° = 1800 °$

Page No 182:

Question 6:

What is the number of diagonals in a
(i) heptagon
(ii) octagon
(iii) polygon of 12 sides?

Answer:

Number of diagonal in an n-sided polygon = $\frac{n (n - 3)}{2}$
(i) For a heptagon:

$n = 7 \Rightarrow \frac{n (n - 3)}{2} = \frac{7 (7 - 3)}{2} = \frac{28}{2} = 14$

(ii) For an octagon:

$n = 8 \Rightarrow \frac{n (n - 3)}{2} = \frac{8 (8 - 3)}{2} = \frac{40}{2} = 20$

(iii) For a 12-sided polygon:

$n = 12 \Rightarrow \frac{n (n - 3)}{2} = \frac{12 (12 - 3)}{2} = \frac{108}{2} = 54$

Page No 182:

Question 7:

Find the number of sides of a regular polygon whose each exterior angle measures:
(i) 40°
(ii) 36°
(iii) 72°
(iv) 30°

Answer:

Sum of all the exterior angles of a regular polygon is $360^{o}$ .

(i)
$Each exterior angle = 40^{o} Number of sides of the regular polygon = \frac{360}{40} = 9$

(ii)
$Each exterior angle = 36^{o} Number of sides of the regular polygon = \frac{360}{36} = 10$

(iii)
$Each exterior angle = 72^{o} Number of sides of the regular polygon = \frac{360}{72} = 5$

(iv)
$Each exterior angle = 30^{o} Number of sides of the regular polygon = \frac{360}{30} = 12$

Page No 182:

Question 8:

In the given figure, find the angle measure x.

Answer:

Sum of all the interior angles of an n-sided polygon = $(n - 2) \times 180 °$

$m ∠ A D C = 180 - 50 = 130^{o} m ∠ D A B = 180 - 115 = 65^{o} m ∠ B C D = 180 - 90 = 90^{o} m ∠ A D C + m ∠ D A B + m ∠ B C D + m ∠ A B C = (n - 2) \times 180 ° = (4 - 2) \times 180 ° = 2 \times 180 ° = 360 ° \Rightarrow m ∠ A D C + m ∠ D A B + m ∠ B C D + m ∠ A B C = 360 ° \Rightarrow 130^{o} + 65^{o} + 90^{o} + m ∠ A B C = 360 ° \Rightarrow 285^{o} + m ∠ A B C = 360^{o} \Rightarrow m ∠ A B C = 75^{o} \Rightarrow m ∠ C B F = 180 - 75 = 105^{o}$
∴ x = 105

Page No 182:

Question 9:

Find the angle measure x in the given figure.

Answer:

For a regular n-sided polygon:
Each interior angle = $180 - (\frac{360}{n})$
In the given figure:
$n = 5 x ° = 180 - \frac{360}{5} = 180 - 72 = 108^{o}$
∴ x = 108

Page No 182:

Question 1:

Tick (✓) the correct answer:
How many diagonals are there in a pentagon?
(a) 5
(b) 7
(c) 6
(d) 10

Answer:

(a) 5

For a pentagon:
$n = 5$

$Number of diagonals = \frac{n (n - 3)}{2} = \frac{5 (5 - 3)}{2} = 5$

Page No 182:

Question 2:

Tick (✓) the correct answer:
How many diagonals are there in a hexagon?
(a) 6
(b) 8
(c) 9
(d) 10

Answer:

(c) 9
Number of diagonals in an n-sided polygon = $\frac{n (n - 3)}{2}$
For a hexagon:

$n = 6 ∴ \frac{n (n - 3)}{2} = \frac{6 (6 - 3)}{2} = \frac{18}{2} = 9$

Page No 182:

Question 3:

Tick (✓) the correct answer:
How many diagonals are there in an octagon?
(a) 8
(b) 16
(c) 18
(d) 20

Answer:

(d) 20

For a regular n-sided polygon:
Number of diagonals =: $\frac{n (n - 3)}{2}$
For an octagon:

$n = 8 \frac{8 (8 - 3)}{2} = \frac{40}{2} = 20$

Page No 182:

Question 4:

Tick (✓) the correct answer:
How many diagonals are there in a polygon having 12 sides?
(a) 12
(b) 24
(c) 36
(d) 54

Answer:

(d) 54
For an n-sided polygon:
Number of diagonals = $\frac{n (n - 3)}{2}$

$∴ n = 12 \Rightarrow \frac{12 (12 - 3)}{2} = 54$

Page No 182:

Question 5:

Tick (✓) the correct answer:
A polygon has 27 diagonals. How many sides does it have?
(a) 7
(b) 8
(c) 9
(d) 12

Answer:

(c) 9

$\frac{n (n - 3)}{2} = 27 \Rightarrow n (n - 3) = 54 \Rightarrow n^{2} - 3 n - 54 = 0 \Rightarrow n^{2} - 9 n + 6 n - 54 = 0 \Rightarrow n (n - 9) + 6 (n - 9) = 0 \Rightarrow n = - 6 o r n = 9 Number of sides cannot be negative . ∴ n = 9$

Page No 183:

Question 6:

Tick (✓) the correct answer:
The angles of a pentagon are x°, (x + 20)°, (x + 40)°, (x + 60)° and (x + 80)°. The smallest angle of the pentagon is
(a) 75°
(b) 68°
(c) 78°
(d) 85°

Answer:

(b) 68°
Sum of all the interior angles of a polygon with n sides = $(n - 2) \times 180 °$

$∴ (5 - 2) \times 180^{o} = x + x + 20 + x + 40 + x + 60 + x + 80 \Rightarrow 540 = 5 x + 200 \Rightarrow 5 x = 340 \Rightarrow x = 68^{o}$

Page No 183:

Question 7:

Tick (✓) the correct answer:
The measure of each exterior angle of a regular polygon is 40°. How many sides does it have?
(a) 8
(b) 9
(c) 6
(d) 10

Answer:

(b) 9
$Each exterior angle of a regular n - sided polygon = \frac{360}{n} = 40 \Rightarrow n = \frac{360}{40} = 9$

Page No 183:

Question 8:

Tick (✓) the correct answer:
Each interior angle of a polygon is 108°. How many sides does it have?
(a) 8
(b) 6
(c) 5
(d) 7

Answer:

(c) 5
Each interior angle for a regular n-sided polygon = $180 - (\frac{360}{n})$

$180 - (\frac{360}{n}) = 108 \Rightarrow (\frac{360}{n}) = 72 \Rightarrow n = \frac{360}{72} = 5$

Page No 183:

Question 9:

Tick (✓) the correct answer:
Each interior angle of a polygon is 135°. How many sides does it have?
(a) 8
(b) 7
(c) 6
(d) 10

Answer:

(a) 8
$Each interior angle of a regular polygon with n sides = 180 - (\frac{360}{n}) \Rightarrow 180 - (\frac{360}{n}) = 135 \Rightarrow \frac{360}{n} = 45 \Rightarrow n = 8$

Page No 183:

Question 10:

Tick (✓) the correct answer:
In a regular polygon, each interior angle is thrice the exterior angle. The number os sides of the polygon is
(a) 6
(b) 8
(c) 10
(d) 12

Answer:

(b) 8
For a regular polygon with n sides:
Each exterior angle = $\frac{360}{n}$
Each interior angle = $180 - \frac{360}{n}$

$∴ 180 - \frac{360}{n} = 3 (\frac{360}{n}) \Rightarrow 180 = 4 (\frac{360}{n}) \Rightarrow n = \frac{4 \times 360}{180} = 8$

Page No 183:

Question 11:

Tick (✓) the correct answer:
Each interior angle of a regular decagon is
(a) 60°
(b) 120°
(c) 144°
(d) 180°

Answer:

Page No 183:

Question 12:

Tick (✓) the correct answer:
The sum of all interior angles of a hexagon is
(a) 6 right ∠s
(b) 8 right ∠s
(c) 9 right ∠s
(d) 12 right ∠s

Answer:

(b) $8 r i g h t ∠ s$
Sum of all the interior angles of a hexagon is $(2 n - 4)$ right angles.
For a hexagon:
$n = 6 \Rightarrow (2 n - 4) right ∠s = (12 - 4) right ∠s = 8 right ∠s$

Page No 183:

Question 13:

Tick (✓) the correct answer:
The sum of all interior angles of a regular polygon is 1080°. What is the measure of each of its interior angles?
(a) 135°
(b) 120°
(c) 156°
(d) 144°

Answer:

(a) 135°

$(2 n - 4) \times 90 = 1080 (2 n - 4) = 12 2 n = 16 o r n = 8 Each interior angle = 180 - \frac{360}{n} = 180 - \frac{360}{8} = 180 - 45 = 135^{o}$

Page No 183:

Question 14:

Tick (✓) the correct answer:
The interior angle of a regular polygon exceeds its exterior angle by 108°. How many sides does the polygon have?
(a) 16
(b) 14
(c) 12
(d) 10

Answer:

(d) 10

$Each exterior angle of a regular polygon = \frac{360}{n} Each interior angle of a regular polygon = 180 - \frac{360}{n} 180 - \frac{360}{n} - 108 = \frac{360}{n} \frac{720}{n} = 180 - 108 = 72 n = \frac{720}{72} = 10$

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