1. Find x in the following figures.

a)

Solution:

125° + m = 180° [Linear pair]

m = 180° – 125° = 55°

125° + n = 180° [Linear pair]

n = 180° – 125° = 55°

we know that exterior angle of a triangle is equal to the sum of 2 opposite interior 2 angles

x = m + n

x = 55° + 55° = 110°

b)

Two interior angles are right angles = 90°

70° + m = 180° [Linear pair]

m = 180° – 70° = 110°

60° + n = 180° [Linear pair]

n = 180° – 60° = 120°

 The given figure is having five sides and is a pentagon.

Thus, sum of the angles of pentagon = 540°

 90° + 90° + 110° + 120° + y = 540°

410° + y = 540° ⇒ y = 540° – 410° = 130°

x + y = 180° (Linear pair)

x + 130° = 180°

x = 180° – 130° = 50°

2. Find the measure of each exterior angle of a regular polygon of

(i) 9 sides (ii) 15 sides

Solution:

We know that sum of angles a regular polygon having side n = (n-2)×180°

(i)here n = 9

Sum of angles a regular polygon having side 9 = (9-2)×180°= 7×180° = 1260°

Each interior angle=1260/9 = 140°

Each exterior angle = 180° – 140° = 40°

(ii) here n =15

 Sum of angles a regular polygon having side 15 = (15-2)×180°

13×180° = 2340°

Each interior angle = 2340/15 = 156°

Each exterior angle = 180° – 156° = 24°

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution:

Each exterior angle = sum of exterior angles/Number of angles

24°= 360°/ Number of sides

Number of sides = 360°/24 = 15

Hence, the regular polygon has 15 sides.

4. How many sides does a regular polygon have if each of its interior angles is 165°?

Solution:

Given, Interior angle = 165°

So,

Exterior angle = 180° – 165° = 15°

Number of sides = sum of exterior angles/ exterior angles

Number of sides = 360/15 = 24

Hence, the regular polygon has 24 sides.

5.

a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

Solution:

Given, Exterior angle = 22°

Now,

Number of sides = sum of exterior angles/ exterior angle

Number of sides = 360/22 = 16.36

No, we can’t have a regular polygon with each exterior angle as 22° because it is not divisor of 360°.

b) Can it be an interior angle of a regular polygon? Why?

Solution:

Interior angle = 22°

Exterior angle = 180° – 22°= 158°

No, we can’t have a regular polygon with each exterior angle as 158° because it is not divisor of 360°.

6.

a) What is the minimum interior angle possible for a regular polygon? Why?

Solution:

As we know that Equilateral triangle is regular polygon which can be constructed with 3 sides at least. Since, sum of interior angles of a triangle = 180°

Hence,

Sum of all interior angles of a regular polygon of side n = (n-2) x 180°

The measure of each interior angle = ((n-2) x 180°)/n

For minimum possible interior angle:

((n-2) x 180°)/n > 0

n>0

Hence, the minimum measure of the angle of an equilateral triangle (n=3)=60°.

b) What is the maximum exterior angle possible for a regular polygon?

Solution:

As we know that Equilateral triangle is regular polygon with 3 sides.

From the above (part a), it conclude that the maximum exterior angle of a regular polygon = 180° – 60° = 120°