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(i) In Regular Polygon of 10 sides, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of polygon of 10 sides is 

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

(10 – 2) x 180°= 1440° . 

Each interior angle = 1440/10 

= 144° 

(ii) In Regular Polygon of 15 sides, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of polygon of 10 sides is 

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

(15 – 2) x 180°= 2340° . 

Each interior angle = 2340/15 

= 156°

In a regular polygon having 10 sides, all sides are same size and measure of all interior

Angles are

Same.

The sum of interior angles of polygon is

(n-2) × 180^o where n is the number of sides of polygon.

(10-2) × 180^o =1440 ^o here n=10

Each interior angle= 1440/10 = 144^o

(ii) In a regular polygon, all sides are same size and measure of all interior angles are

Same.

The sum of interior angles of polygon of 15 sides is

(n-2) × 180^o where n is the number of sides of polygon.

(15-2) × 180^o =1440^o here n=15

Each interior angle= 2340/15 = 156^o

When one side is increased, the angle is increased by 180°.

∴ Sum of the angles of a 13 sides Polygon = 1800° + 180 = 1980°

Sum of angles of a polygon with 20 sides. 

= (20 – 2) × 180

= 18 × 180

Each angle = \(\frac{18\times180}{20}\) = 162°

(a) Sum of angles = (12 – 2 ) × 180 1800°

(b) Sum of angles = (15 – 2) × 180 = 2340°

(c) Sum of angles = (20 – 2) × 180 = 3240°

(d) Sum of angles = (24 – 2) × 180 = 3960°

(i) No. of Sides = 360°/Exterior Angle 

= 360/40 

= 9 

Number of sides is 9 of regular polygon whose exterior angle is 40°.

(ii) No. of Sides = 360° / Exterior Angle 

= 360/36 

= 10 

Number of sides is 10 of regular polygon whose exterior angle is 36°.

(iii) No. of Sides = 360° / Exterior Angle 

= 360/72 

= 5 

Number of sides is 5 of regular polygon whose exterior angle is 72°.

(iv) No. of Sides = 360° / Exterior Angle 

= 360/30 

= 12 

Number of sides is 12 of regular polygon whose exterior angle is 30°.

• Measurement of a side and angle between two adjacent sides.
• diagonals.

∵ BEST is a parallelogram. 

∴ ∠B + x° = 180° 

Adjacent angles of a parallelogram are supplementary 

⇒ 110° + x° = 180° 

⇒ x° = 180° – 110° 

⇒ x° = 70° 

⇒ x = 70 

∵ Opposite angles of a parallelogram are equal 

∴ x° = y° 

⇒ 70° = y° 

⇒ y = 70 and z° = 110° 

⇒ z = 110

The value of each angle in a rectangle is 90°.

∵ PQ || SR
∴ x + 90° = 180°
⇒ x = 180° – 90° = 90°
Again ∴ PQRS is a quadrilateral
∴ x + y + 130° + 90° = 360°
∵ We know that sum of all angles of a quadrilateral = 360°
⇒ 90° + y + 130° + 90° = 360°
⇒ y + 310° = 360°
⇒ y = 360° – 310°
⇒ y = 50°

A trapezium cannot have all angles equal as its opposite sides become parallel. But trapezium is a quadrilateral with one pair of parallel sides. A trapezium cannot have all sides equal as again its opposite sides become parallel. But trapezium is a quadrilateral with one pair of parallel sides.

We know that the diagonals of parallelogram bisect each other
∴ x + 3 = 20 ….(1)
y = 16 ……(2)
From equation (1)
x = 20 – 3 = 17
y = 16

To make a concrete rectangular slab the man son should ensure

• its opposite sides equal;
• its diagonals equal;
• each angle measure 90°.

Opposite sides.

Since opposite sides of a parallelogram are equal in lengths, therefore

3x + 1 = 16 

⇒ 3x = 16 – 1

⇒ 3x = 15 

⇒ x = 5 cm.

5y – 6 = 24 

⇒ 5y = 24 + 6

⇒ 5y = 30 

⇒ y = 6 cm.

A rectangle is a parallelogram with equal angles.

In parallelogram PQ = SR
so 3x – 1 = 29
or 3x = 29 + 1
or 3x = 30
or x = 30/3
= 10 cm
In parallelogram PS = QR
so 4y = 24
or y = 24/4
= 6 cm
so we can write x = 10 cm and y = 6 cm

(b) quadrilaterals

By showing given parallelogram RING,
∵∠R = 70° (given that)
∴∠N = 70°
∵∠N, ∠R are opposite angles
we know that the sum of any two adjacent angles of a parallelogram is 180°, therefore ∠I = 180° – 70° = 110°
so we can write ∠G = 110° 

∵ ∠I, ∠G are opposite angles.

Square is a regular quadrilateral.

(c) 4 cm and 5 cm

(b) 90°, 90°, 90°, 90°

(i) 3 sides

Regular Polygon: A regular polygon is an enclosed figure. In a regular polygon minimum sides are three.

(ii) 4 sides

A regular polygon with 4 sides is known as quadrilateral.

(iii) 6 sides

A regular polygon with 6 sides is known as hexagon.

(i) In Regular Pentagon, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of pentagon is 

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

(5 – 2) × 180°= 540°.

Each interior angle = 540/5 = 108°

As, we know that Sum of Interior Angle and Exterior Angle is 180° 

Exterior Angle + Interior Angle = 180° 

Exterior Angle +108° =180° 

So, Exterior Angle = 180°- 108° 

= 72°

(ii) In Regular Hexagon, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of hexagon is 

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

(6 – 2) × 180°= 720° 

Each interior angle = 720/6 = 120° 

As, we know that Sum of Interior Angle and Exterior Angle is 180° 

Exterior Angle + Interior Angle = 180° 

Exterior Angle + 120° = 180° 

So, Exterior Angle = 180°- 120° 

= 60°

(iii) In Regular Heptagon, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of heptagon is 

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

(7 – 2) X 180°= 900°.

Each interior angle = 900/7 = 128.57°

As, we know that Sum of Interior Angle and Exterior Angle is 180° 

Exterior Angle + Interior Angle = 180° 

Exterior Angle + 128.57° =180° 

So, Exterior Angle = 180°– 128.57° 

= 51.43°

(iv) In Regular Decagon, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of decagon is 

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

(10 – 2) X 180°= 1440° . 

Each interior angle = 1440/10 = 144°

As, we know that Sum of Interior Angle and Exterior Angle is 180° 

Exterior Angle + Interior Angle = 180° 

Exterior Angle + 144° =180° 

So, Exterior Angle = 180°- 144° 

= 36°

(v) In Regular Polygon of 15 sides, all sides are of same size and measure of all interior angles are same. 

The sum of interior angles of polygon of 15 sides is 

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

(15 – 2) X 180°= 2340°. 

Each interior angle = 2340/15 = 156°

As, we know that Sum of Interior Angle and Exterior Angle is 180° 

Exterior Angle + Interior Angle = 180° 

Exterior Angle + 156° = 180° 

So, Exterior Angle = 180°- 156° 

= 24°

Third angle = 180 – (40 + 60) = 80°

Exterior angles = 180 – 40, 180 – 60,

180 – 80, 140, 

ie, 140°, 120° and 100°

4^th angle = 360 – (130 + 70 + 60) = 100°

Exterior angles = 120°, 110°, 50°, 80°

(c) 41°

ΔAQD is an equilateral Δ 

⇒ ∠QAD = ∠QDA = ∠AQD = 60° 

∠BAD = 360° – (135° + 90° + 60°) 

= 360° – 285° = 75° (Angles round a pt.) 

Also, in quad. ABCD, 

∠CDA = 360°– (100° + 106° + 75°) 

= 360° – 281° = 79°. 

∴ ∠PDQ =180° – (∠CDA+ ∠QDA)         (CDP is a st. line) 

= 180° – (79° + 60°) = 180° – 139° = 41°.

2x + 30 + 3x – 10 + 100 = 360

5x + 120 = 360

5x = 360 – 120 = 240

∴ x = \(\frac{240}{5}\) = 48

\(\frac{360}{13}\) is a fraction. So it is not possible.

(i) Simple curve

Fig (i), (ii), (v), (vi) and (vii) are simple curves.

(ii) Simple closed curve

Fig (i), (ii), (v), (vi) and (vii) are simple closed curves.

(iii) Polygon

Fig (i) and (ii) are polygons. Polygons are minimum three sided enclosed figure.

(iv) Convex polygon

Fig (ii) is a convex polygon. In a convex polygon all the vertices are pointing outwards.

(v) Concave polygon

Fig (i) is a concave polygon. In a concave polygon all the vertices are not pointing outwards.

(vi) Not a curve

Fig (viii) is not a curve.

Closed curves: (ii), (iii) and (vi)

Open curves: (i), (iv) and (v)

(c) 45°

∠DCB = 90° (ABCD is a square) 

∠TCB = 60° (DCT is an equilateral Δ) 

∴ ∠DCT = 90° + 60° = 150° 

DC = CB (Adj sides of a square) 

CB = CT (Sides of an equilateral Δ) 

⇒ DC = CT ⇒ ∠CTD =∠CDT (isos. Δ property) 

In ΔDCT, ∠CTD = \(\frac12\) (180° –∠DCT) 

= \(\frac12\) (180° – 150°) = 15° 

∴ ∠BTD =∠BTC – ∠CTD= 60° – 15° = 45°.

One angle of a regular pentagon = 108° 

One angle of a regular hexagon = 120°.

One angle of a square = 90°

∴ ∠BAC = 360 – (108 + 20 + 90)

360 – 318 = 42°.

Exterior angle = 180 – 168 = 12°

Sum of exterior angles = 360°

∴ Number of sides = \(\frac{360}{12}\) = 30

Sum of angles of a regular pentagon = (5 – 2) 180°

= 3 × 180 = 540°

One angle of a regular pentagon = \(\frac{540}{5}\) = 108°

Sum of angles of a regular hexagon = (6 – 2) 180°

= 4 × 180 = 720°

One of its angle = \(\frac{720}{6}\) = 120°

∴ ∠PQR = 360 – (108 + 120)

= 360 – 228 = 132°

If the measure of each interior angle of a polygon is less than 180°, then it is called a convex polygon.

If the measure of at least one interior angle of a polygon is greater than 180°, then it is a concave or rentrant polygon.

Sum of exterior angles = 360°

One of the exterior angle = 6°

Number of sides = \(\frac{360}{6}\) = 60, yes we can draw; 

If one exterior angle is 7°

Number of sides = \(\frac{360}{7}\) = 51.42

Not a whole number. The polygon cannot be drawn.

A polygon with all sides and all angles equal is called a regular polygon.

(b) \(51\frac{3°}{7}\)

Exterior angle = \(\frac{360°}{\text{number of sides}}\) = \(\frac{360°}{7} = \) \(51\frac{3°}{7}\).

Properties of polygons : For a polygon of n sides, 

(i) Sum of interior angles = (2n – 4) × 90° 

(ii) Sum of exterior angles= 360° (always) 

(iii) Each interior angle = \(\frac{(2n-4)}{n}\) x 90° (regular polygon). 

(iv) Each exterior angle \(\frac{360°}{n}\) (in regular polygon) 

(v) Interior angle + exterior angle = 180° (always)

Only figure (ii) and (iii) are polygons.

One angle of a regular hexagon = 120°

∠EFD = ∠EDF = 30°; ∠F = 120°

∴ ∠AFD = 90°

Similarly all the angles of ACDF is 90°

∴ ACDF is a rectangle.

(x + 180°), (x – 180°)

Sum of angles of a regular hexagon = (6 – 2) 180 = 720°

= 4 × 180 = 720°

One angle = \(\frac{720}{6}\) = 120°

consider ∆ EFO

∠E = 120°

∠EFD = ∠EDF = 30°(Angles opposite to equal sides of an isosceles triangle are equal) 

Similarly ∠AFB = 30° 

∴∠DFB = 120 – (30 + 30) = 60° 

∴ ∠FBD = 60°, 

∠FDB = 60° 

∴ ∆ FDB is an equilateral triangle.

Let the number of sides be n 

Sum of all exterior angle of n side regular polygon = 360° 

∴ each exterior angle = 360°/n

According to question 45° = 360°/n

⇒ 45° n = 360° 

⇒ n = 360°/45° = 8

⇒ n = 8 

Number of sides = 8

Here n = 7 

∴ Sum of all interior angles of a regular polygon 

= (n – 2) 180° 

= (7 – 2) 180° 

= 5 x 180° 

= 900°

Regular polygon—A closed figure of three or more than three equal sides is called regular polygon. 

(i) Closed figure of 5 sides : pentagon. 

(ii) Closed figure of 6 sides : hexagon. 

(iii) Closed figure of 8 sides : Octagon.

We know sum of all exterior angles of a regular polygon = 360°

 ⇒ 110° + 115° + x° = 360° 

⇒ 225° + x° = 360° 

⇒ x° = 360° – 225° 

⇒ x° = 135°

Sum of interior angles of a hexagon 

= (n – 2) x 180° 

= (6 – 2) x 180°

= 4 x 180° = 720° 

According to question, 

⇒ 165° + x° + x° + x° + x° + x° = 720° 

⇒ 165° + 5x° = 720° 

⇒ 5x° = 720° – 165° 

⇒ 5x° = 555° 

⇒ 5x = 555 

⇒ x = 555/5 

⇒ x = 111