PROBLEMS ON INTERIOR AND EXTERIOR ANGLES OF POLYGON

Problem 1 :

The number of sides of a regular polygon where each exterior angle has a measure of 45^º is ……

a)  8        b) 10       c) 4     d) 6

Solution :

Sum of the exterior angles of a regular polygon = 360 ^º

Exterior angle has a measure = 45^º

Number of sides of a regular polygon = 360/45

= 8

Number of sides of a regular polygon is 8 sides.

Hence, option a. is correct.

Problem 2 :

a)  60^º      b) 140^º      c) 150^º     d) 108^º

Solution :

A regular pentagon has all its five sides equal and all five angles are also equal.

S = [(n – 2) × 180^º]/n

= [(5 – 2) × 180^º]/5

= (3 × 180^º)/5

= 540^º/5

= 108^º

Hence, option d. is correct.

Problem 3 :

If two adjacent angles of a parallelogram are in the ratio 2 : 3, then the measure of angles are 

(a) 72^º, 108^º     (b) 36^º, 54^º     (c) 80^º, 120^º    (d) 96^º, 144^º

Solution :

Let the two adjacent angles of a parallelogram are in the ratio be 2x and 3x.

So, 2x + 3x = 180^º

[Interior angles on the same side of transversal].

5x = 180^º

Dividing 5 on both sides.

5x/5 = 180^º/5

x = 36^º

Therefore the measure of angles are, 

2 × 36^º = 72^º

2 × 36^º = 108^º

Hence, option a. is correct.

Problem 4 :

If PQRS is a parallelogram, then ∠P - ∠R is equal to

a) 60^º     b) 90^º    c) 80^º    d) 0^º

Solution :

Opposite angles are equal to each other in parallelogram.

Given, ∠P - ∠R

∠P = ∠R

∠R - ∠R = 0^º

Hence, option d. is correct.

Problem 5 :

The sum of adjacent angles of a parallelogram is

a) 180^º     b) 120^º  c) 360^º  d) 90^º

Solution :

The sum of adjacent angles of a parallelogram is 180^º. Because both angles are co – interior angles.

Hence, option a. is correct.

Problem 6 :

The number of sides of a regular polygon whose each interior angle is of 135^º is ……

a) 6       b) 7      c) 8        d) 9

Solution :

Sum of colinear interior and exterior angle is 180^º.

Interior angle + exterior angle = 180^º

135^º + exterior angle = 180^º

Exterior angle = 180^º - 135^º

Exterior angle = 45^º

Number of sides of a regular polygon = 360^º/Exterior angle

= 360^º/45^º

= 8

So, the number of sides of a regular polygon is 8.

Hence, option c. is correct.

Problem 7 :

The figure given below is a regular pentagon, find the value of x.

Solution :

Number of sides of pentagon = (n - 2) 180/n

Here n = 5

= (5 - 2) 180/5

= 3(36)

= 108

One interior angle = 108

Since it is regular pentagon, all sides will be equal and all interior angles will be equal. Then in the triangle we see at the top, the sum of interior angles of triangle is 180.

108 + y + y = 180

108 + 2y = 180

2y = 180 - 108

2y = 72

y = 72/2

y = 36

So, the value of y is 36 degree.

Problem 8 :

A polygon has an interior angle that is five times larger than the exterior angle. How many sides does it have?

Solution :

Interior angle + exterior angle = 180

Let x be the exterior angle, then interior angle = 5x

5x + x = 180

6x = 180

x = 180/6

x = 30

interior angle = 180 - 30

= 150

(n - 2) 180/n = 150

180n - 360 = 150n

180n - 150n = 360

30n = 360

n = 360/30

n = 12

So, the required number of sides is 12.

Problem 9 :

The diagram shows three regular pentagons meeting at a point. Work out the size of the angle marked x. You must show all your working.

Solution :

One interior angle measure of pentagon = 108

3(interior angle of pentagon) + x = 360

3(108) + x = 360

324 + x = 360

x = 360 - 324

x = 36

So, the value of x is 36 degree.

Problem 10 :

The diagram shows regular 5 sided polygon with center O.

Solution :

One interior angle of pentagon = 108

Angle at the center = 360 / 5

= 72

x + x + 72 = 180

2x + 72 = 180

2x = 180 - 72

2x = 108

x = 108/2

x = 54

2x + y = 180

2(54) + y = 180

108 + y = 180

y = 180 - 108

y = 72

Problem 11 :

Shape A is a regular triangle. Shape B is a regular octagon.

Another regular polygon, P, is shown on the diagram.

How many sides does polygon P have? You must show your working.

Solution :

Number of sides of octagon = 8

One interior angle of octagon = (n - 2)(180/n)

= (8 - 2)(180/8)

= 6(22.5)

= 135

Since shape A is regular polygon which has three sides and one interior angle measure is 60 degree.

One interior angle = 60

Interior angle of octagon + interior angle of triangle + interior angle of polygon A

= 360

135 + 60 + interior angle of polygon A = 360

195 + interior angle of polygon A = 360

interior angle of polygon A = 360 - 195

interior angle of polygon A = 165

(n - 2) (180/n) = 165

180n - 360 = 165n

180n - 165n = 360

15n = 360

n = 360/15

n = 24

So, the number of sides of polygon A is 24.