**In each of the following polygons find in degrees the sum of the interior angles and the sum of exterior angles.**

**(i) Hexagon **

**(ii) Octagon **

**(iii) pentagon **

**(iv) nonagon **

**(v) decagon **

**Solution: **

(i) Hexagon

Numbers of sides, n = 6

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

= (2 x 6 – 4) x 90°

= (12 – 4) 90°

= 8 x 90°

= 720°

Sum of the exterior angles = 360°

(ii) Octagon

Numbers of sides, n = 8

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

= (2 x 8 – 4) x 90°

= (16 – 4) 90°

= 12 x 90°

= 1080°

Sum of the exterior angles = 360°

(iii) Pentagon

Numbers of sides, n = 5

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

= (2 x 5 – 4) x 90°

= (10 – 4) 90°

= 6 x 90°

= 540°

Sum of the exterior angles = 360°

(iv) Nonagon

Numbers of sides, n = 9

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

= (2 x 9 – 4) x 90°

= (18 – 4) 90°

= 14 x 90°

=1260°

Sum of the exterior angles = 360°

(v) Decagon

Numbers of sides, n = 10

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

= (2 x 10 – 4) x 90°

= (20 – 4) 90°

= 16 x 90°

= 1440°

Sum of the exterior angles = 360°

**How many sides dose a polygon have if the sum of the interior angle is**

**(i) 540° **

**(ii) 900° **

**(iii) 1440° **

**(iv) 7 straight angles **

**(v) 8 straight angles **

**Solution:**

(i) Sum of the interior angles = (2n – 4) 90° = 540°

2n – 4 = ^{540}/_{90} = 6

2n = 6 + 4 = 10

n = ^{10}/_{2} = 5

**The polygon has 5 sides**

(ii) Sum of the interior angles = (2n – 4) 90° = 900°

→ 2n – 4 = ^{900}/_{90} = 10

→ 2n = 10 + 4 = 14

→ n = ^{14}/_{2} = 7

**The polygon has 7 sides**

(iii) The Sum of the interior angles = (2n – 4) 90° = 1440°

→ 2n – 4 = ^{1440}/_{90} = 16

→ 2n = 16 + 4 = 20

→ n = ^{20}/_{2 }= 10

**The polygon has 10 sides **

(iv) Sum of the interior angles = (2n – 4) 90° = 7 straight angles = 7 x 180

→ (2n – 4) = ^{7 x 180}/_{90}

→ 2n – 4 = ^{7 x 180}/_{90} = 7 x 2 = 14

→ 2n = 14 + 4 = 18

→ n = ^{18}/_{2} = 9

**The polygon has 9 sides**

(v) Sum of the interior angles = (2n – 4) 90° = 8 right angles

→ (2n – 4) 90° = 8 x 90°

→ 2n – 4 = 8 x 9090 = 8

→ 2n = 8 + 4 = 12

→ n = 122 = 6

The polygon has 6 sides

**Find the measure of each exterior angle of a regular polygon with sides:**

**(i) 40 **

**(ii) 30 **

**(iii) 20 **

**(iv) 18 **

**(v) 16 **

**(vi) 2x **

**(vii) (2a +4b) **

**Solutions: **

(i) Number of sides n = 40

Measure of each exterior angle = ( ^{360}/_{n} ) ° = ^{360}/_{40} = **9° **

(ii) Number of sides n = 30

Measure of each exterior angle = ( ^{360}/_{n} ) ° = ^{360}/_{30} = **12° **

(iii) Number of sides n = 20

Measure of each exterior angle = ( ^{360}/_{n} ) ° = ^{360}/_{20} = **18° **

(iv) Number of sides n = 18

Measure of each exterior angle = ( ^{360}/_{n} ) ° = ^{360}/_{18} = **20° **

(v) Number of sides n = 16

Measure of each exterior angle = ( ^{360}/_{n} ) ° = ^{360}/_{16} = 22.5°

(vi) Number of sides n = 2x

Measure of each exterior angle = ( ^{360}/_{n }) ° = ^{360}/_{2x} = **( **^{𝟏𝟖𝟎}/_{𝐱} **)°**

** **

(vii) Number of sides n = 2a + 4b

Measure of each exterior angle = ( ^{360}/_{n} ) ° = ^{360}/_{2a + 4b} = **( **^{𝟏𝟖𝟎}/_{𝐚 + 𝟐𝐛} **)° **

** **

**Find the numbers of sides of regular polygon. If each exterior angle measures.**

**(i) 10° **

**(ii) 20° **

**(iii) 30° **

**(iv) 40° **

**(v) 45° **

**(vi) 60° **

**(vii) 72° **

**(viii) 120° **

Solution:

**Solution: **

(i) Measure of each exterior angle = x° = 10°

Number of sides = ^{360}/_{x }= ^{360}/_{10} = **35 **

(ii) Measure of each exterior angle = x° = 20°

Number of sides = ^{360}/_{x} = ^{360}/_{20} = **18**

(iii) Measure of each exterior angle = x° = 30°

Number of sides = ^{360}/_{x} = ^{360}/_{30} = **12 **

(iv) Measure of each exterior angle = x° = 40°

Number of sides = ^{360}/_{x} = ^{360}/_{40} = **9**

v) Measure of each exterior angle = x° = 45°

Number of sides = ^{360}/_{x} = ^{360}/_{45} = **8**

(vi) Measure of each exterior angle = x° = 60°

Number of sides = ^{360}/_{x} = ^{360}/_{60} = **6**

(vii) Measure of each exterior angle = x° = 72°

Number of sides = ^{360}/_{x} = ^{360}/_{72} = **5**

(viii) Measure of each exterior angle = x° = 120°

Number of sides = ^{360}/_{x} = ^{360}/_{120} = **3**

**Find the numbers of sides of a regular polygon it each exterior is equal to**

**(i) Its adjacent interior angle. **

**(ii) Twice its adjacent interior angle. **

**(iii) Half its adjacent interior angle. **

**(iv) One-third of its adjacent interior angle. **

**Solution:**

(i) Exterior angle + its adjacent interior = 180°

e + i = 180°

e = i

e + e = 180°

2e = 180°

e = ^{180}/_{2} = 90°

Number of sides = ^{360}/_{e} = ^{360}/_{90} = **4 **

(ii) e + i = 180°

e = ^{2}/_{i}

i = ^{e}/_{2}

e + ^{e}/_{2} = 180°

^{2e+e}/_{2} = 180°

3e = 180 x 2 = 360°

e = ^{360}/_{3} = 120°

Number of sides = ^{360}/_{e} = ^{360}/_{120} = **3**

(iii) e + i = 180°

e = ^{i}/_{2}

i = 2e

e + 2e= 180°

3e = 180°

e = ^{180}/_{3} = 60°

Number of sides = ^{360}/_{e} = ^{360}/_{60} = **6**

(iv) e + i = 180°

e = ^{i}/_{3 }

i = 3e

e + 3e= 180°

4e = 180°

e = ^{180}/_{4} = 45°

Number of sides = ^{360}/_{e} = ^{360}/_{45} = **8**

**find the number of sides in a regular polygon if each interior angle is**

**(i) twice its adjacent exterior angle **

**(ii) Four times the adjacent exterior angle. **

**(iii) Eight times the adjacent exterior angle. **

**(iv) Seventeen times the adjacent exterior angle. **

**Solution:**

(i) e + i = 180°

i = 2e

e + 2e = 180°

3e = 180°

e = ^{180}/_{3} = 60°

Number of sides = ^{360}/_{e} = ^{360}/_{60} **= **6

(ii) e + i = 180°

i = 4e

e + 4e = 180°

5e = 180°

e = ^{180}/_{5} = 36°

Number of sides = ^{360}/_{e} = ^{360}/_{36} **= **10

(iii) e + i = 180°

i = 8e

e + 8e = 180°

9e = 180°

e = ^{180}/_{9} = 20°

Number of sides = ^{360}/_{e} = ^{360}/_{20} **= **18

(iv) Seventeen times the adjacent exterior angle

e + i = 180°

i = 17e

e + 17e = 180°

18e = 180°

e = ^{180}/_{18} = 10°

Number of sides = ^{360}/_{e} = ^{360}/_{10} **= **36

**The angles of a convex polygon are in the ratio 2 : 3 : 5 : 9 :11. Find the measure of each angle.**

**Solution: **

Ratio of the angle = 2 : 3 : 5 : 9 :11.

Let the angle be 2x, 3x, 5x, 9x and 11x.

The polygon has five angles and therefore five sides.

Sum of angle = (2n – 4)90°

= (2 x 5 – 4)90°

= (10 – 4)90

= 6 x 90°

2x + 3x + 5x + 9x +11x = 6 x 90°

30x = 6 x 90°

x = ^{6 x 90}/_{30} = 6 x 3 = 18°

The angles are 2x = 2 x 18 = 36°

3x = 3 x 18 = 54°

5x = 5 x 18 = 90°

9x = 9 x 18 = 162°

11x = 11 x 18 = 198°

**Prove that the opposite sides of a regular hexagon are parallel.**

**Solution:**

Construction: in the regular hexagon ABCDEF, join FC. In the quadrilateral ABCF, AF = BC and ∠FAB = ∠CBA = 120°

ABCF is an isosceles trapezium

FC | | AB ……. (1)

Similarly ED | | FC ……. (2)

From (1) and (2) AB | | ED

Similarly BC | | FE and CD | | FA

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