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(i) Interior angles are— a, b, c, d, e 

(ii) Exterior angles are— p, q, r, s, t

Each interior angle = 165° 

∴ Each exterior angle = 180° – 165° = 15° 

Let the number of sides be n. 

Sum of all exterior angles of polygon = 360° 

∴ Value of each angle = 360°/n 

According to question = 360°/n = 15° 

⇒ n = 360°/15° = 24

Therefore, the number of sides be 24

One angle of a pentagon = 120°

Let remaining four angles be x, x, x and x

Their sum = 4x + 120°

But sum of all the interior angles of a pentagon = (2n – 4) x 90°

= (2 x 5 – 4) x 90° = 540°

= 3 x 180° = 540°

∴ 4x + 120° = 540°

4x = 540° – 120°

4x = 420

x = (420/4)° ⇒ x = 105°

∴Equal angles are 105° (Each)

The sum of interior angles of hexagon is 

= (n – 2) × 180° [n is number of sides of polygon)] 

= (6 – 2) X 180° 

= 720° .

1 right \(\angle\) s = 90°

So, 720°= 8 right \(\angle\) s

This is a regular pentagon, as all sides are of equal length. 

AB = BC = CD = DE = EA 

The sum of interior angles of poligon is 

= (n – 2) × 180° [n is number of sides of polygon)] 

= (5 – 2) X 180° [ for pentagon n = 5] 

= 540° 

Since, it is a regular pentagon. It’s all interior angle will be equal. 

Size of Interior Angle x = 540/5 

= 108°

Number of diagonals in Pentagon is

\(=n\times\frac{n\,-\,3}{2}\) [n represents number of sides]

\(=5\times\frac{5\,-\,3}{2}\)

\(=5\times\frac{2}{2}\)

= 5 

So, Number of diagonals in pentagon is 5.

(a) 5

Explanation:

We know that to calculate number of diagonals in pentagon is

n × (n -3)/2

But here n=5

5 × (5-3)/2 =5

(c) 9

Explanation:

We know that to calculate number of diagonals in hexagon is

n × (n -3)/2

But here n=6

6 × (6-3)/2 = 9

7 diagonals 8 triangles.

(c) 9

Explanation:

We know that to calculate number of diagonals in octagon is

Number of diagonals is= n × (n -3)/2

27= n × (n -3)/2

n (n-3) = 54

n²– 3n = 54

n²– 3n-54 = 0

(n + 6) (n – 9)=0

n=-6 or n=9

So that we are calculating the sides it should be positive, therefore sides of polygon has 27 diagonals is 9

(d) 54

Explanation:

We know that to calculate number of diagonals in octagon is

n × (n -3)/2

But here n=12

12 × (12-3)/2 = 54

Let ABCD is a parallelogram According to question

∠A : ∠B = 1 : 5

∠A + ∠B = 180°

∵Sum of two adjacent angles of a parallelogram is 180°

Sum of Ratio = 1 + 5 = 6

∴∠A = 1/6 x 180° = 30°

∠B = 5/6 x 80° = 150°

∵ Opposite angles of a parallelogram are equal

∠C = ∠A = 30°

and ∠D = ∠B = 150°

∴Angles of a parallelogram are 30°, 150°, 30° and 150°.

∵ Diagonal of a rectangle bisect each other
∴ SE = SO
and SH = SP
∴ EO = 2SE
= 2 (3x + 1)
and HP = 2 SH = 2 (2x + 4)
∵ Diagonal of a rectangle are equal.
∴ EO = HP
⇒ 2(3x + 1) = 2(2x + 4)
⇒ 3x + 1 = 2x + 4
⇒ 3x – 2x = 4 – 1
⇒ x = 3

(i) ∵ Opposite sides of a parallelogram are equal.

∴3x = 18
and 3y – 1 = 26
⇒ x = 18/3 = 6
and 3y = 26 + 1
⇒ 3y = 27
⇒ y = 27/2 = 9
So, x = 6 cm and y = 9 cm

(ii) ∵ Diagonals of a parallelogram bisect each other

x + y = 16….(1)
and y + 7 = 20….(2)
From equation (2),
y = 20 – 7 = 13
We put y = 13 in equation (1)
x + 13 = 16
=> x = 16 – 13 = 3
So, x = 3 cm and y = 13 cm

Exterior Angle = 40°

No. of Sides = 360 / Exterior Angle

= 360 / 40 

= 9

(b) 9

Explanation:

Given exterior angle= 40°

But we know that Number of sides = 360/ exterior angle

Number of sides = 360/40=9

The sum of exterior angles in a polygon is 360°.

(i) The sum of the interior angles in a triangle is 180°. It is a triangle and has 3 sides.

(ii) If the sum of the interior angles is 720°, the polygon has six sides.

(iii) If the sum of the interior angles is 360°, the polygon is a quadrilateral it has 4 sides.

The exterior angles are equal as all the angles are equal.

Let the interior angle is x, then the exterior angle is 2x.

x + 2x = 3x = 180

∴ x = 60

(i) Interior angles are 60° each and exterior angles are 120° each.

(ii) The polygon has 3 sides. It is a triangle.