PROBLEMS INVOLVING INTERIOR ANGLES OF A POLYGON

Find the value of x in the following problems, give brief reasons for your answers.

Problem 1 :

Solution :

Number of sides of the polygon given above = 5.

Sum of interior angles of given polygon = (n - 2)180°

= (5 - 2)180°

= (3)180°

= 540°

108° + 108° + 108° + 108° + x° = 540°

432 + x = 540

Subtract 432 from both sides.

x = 108

Problem 2 :

Solution :

Number of sides of the polygon given above = 6

Sum of interior angles of given polygon = (n - 2)180°

= (6 - 2)180°

= (4)180°

= 720°

130° + x° + 118° + 112° + 120° + 90° = 720°

570 + x = 720

Subtract 570 from both sides.

x = 150

Problem 3 :

Solution :

Number of sides = 5

Sum of interior angles of given polygon = (n - 2)180°

= (5 - 2)180°

= (3)180°

= 540°

2x° + 2x° + 2x° + 2x° + x° = 540°

9x = 540

Divide both sides by 9.

x = 60

Problem 4 :

Solution :

Number of sides = 6

Sum of interior angles of given polygon = (n - 2)180°

= (6 - 2)180°

= (4)180°

= 720°

120° + 150° + x° + x° + x° = 720°

270 + 3x = 720

Subtract 270 from both sides.

3x = 450

Divide both sides by 3.

x = 150

 

Problem 5 :

Solution :

Number of sides = 7

Sum of interior angles of given polygon = (n - 2)180°

= (7 - 2)180°

= (5)180°

= 900°

60° + x° + x° + x° + x° + 2x° + 90° = 720°

150 + 6x = 720

Subtract 150 from both sides.

6x = 570

Divide both sides by 6.

x = 95

Problem 6 :

A regular polygon has interior angle of 156°. How many sides does the polygon have?

Solution :

Polygon is made up of a finite number of straight lines which are connected to form a closed polygonal circuit.

Sum of interior and exterior angle = 180°

Interior angle = 156°

Exterior angle = 180° - 156° = 24°

Number of sides of polygon = 360°/exterior angle

= 360°/24°

= 15

So, the polygon has 15 sides.

Problem 7 :

Five identical isosceles triangles are put together to form a regular pentagon.

a) Explain why 5x = 360.

b) Find x and y.

Solution :

In each triangle, sum of interior angles = 180°

x° + y° + y° = 180°

x + 2y = 180 -----(1)

From (a)

5x = 360

Divide both sides by 5.

x = 72

Substitute x = 72 in (1).

72 + 2y = 180

Subtract 72 from both sides.

2y = 108

Divide both sides by 2.

y = 54

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