FIND THE MISSING INTERIOR AND EXTERIOR ANGLES OF A POLYGON

Find the missing interior and for exterior angles of a given polygon:

Problem 1 :

Solution :

y + 121° = 180° (linear pair)

y = 180 - 121

y = 59

x° + 84° + 100° + 59° = 360°

x° + 243 = 360°

x° = 360 - 243

x° = 117

So, the missing angle of x is 117°.

Problem 2 :

Solution :

The sum of exterior angles of a polygon is 360°.

x° + 100° + 120° = 360°

x° + 220° = 360°

x° = 360 - 220

x° = 140

So, the missing angle of x is 140°.

Problem 3 :

Solution :

The sum of exterior angles of a polygon is 360°.

x° + 95° + 70° + 90° = 360°

x° + 255° = 360°

x° = 360 - 255

x° = 105

So, the missing angle of x is 105°.

Problem 4 :

Solution :

The sum of exterior angles of a polygon is 360°.

x° + 70° + 70° + 70° + 60° + 40° = 360°

x° + 310° = 360°

x° = 360 - 310

x° = 50

So, the missing angle of x is 50°.

Problem 5 :

Solution :

The sum of exterior angles of a polygon is 360°.

x° + 93° + 55° + 102° = 360°

x° + 250° = 360°

x° = 360 - 250

x° = 110

So, the missing angle of x is 110°.

Problem 6 :

Solution :

The sum of exterior angles of a polygon is 360°.

x° + 159° + 31° + 91° = 360°

x° + 281° = 360°

x° = 360 - 281

x° = 79

So, the missing angle of x is 79°.

Problem 7 :

A dodecagon is a polygon with 12 sides. Find the interior and exterior angles of a regular dodecagon.

Solution :

Number of sides of dodecagon = 12

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

= (12 - 2) 180

= 10(180)

= 1800

Each interior angle = 1800/12

= 150

Exterior angle = 180 - 150

= 30

Problem 8 :

An octadecagon is a polygon with 18 sides. Find the interior and exterior angles of a regular octadecagon.

Solution :

Number of sides of octadecagon = 18

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

= (18 - 2) 180

= 16(180)

= 2880

Each interior angle = 2880/18

= 160

Exterior angle = 180 - 160

= 20

Problem 9 :

An icosagon is a polygon with 20 sides. Find the sum of the interior angles in a regular icosagon.

Solution :

Number of sides of icosagon = 20

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

= (20 - 2) 180

= 18(180)

= 3240

So, the sum of interior angles of icosagon is 3240.

Problem 10 :

The size of each interior angle in a regular polygon is 135°. How many sides does the regular polygon have?

Solution :

Interior angle of polygon = 135

(n - 2)180/n = 135

(n - 2) 180 = 135n

180n - 360 = 135n 

180n - 135n = 360

45n = 360

n = 360/45

n = 8

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

Problem 11 :

Pictured below is a regular nonagon.

(a) Find the size of angle x. (b) Find the size of angle y.

Solution :

Number of sides of the polygon above = 9

Measure of one interior angle = (9 - 2) (180/9)

= 7(20)

= 140

So, the value of y is 140 degree.

Sum of interior angle and exterior angle = 180

140 + x = 180

x = 180 - 140

x = 40

So, the value of x is 40.

Problem 12 :

Pictured below is part of a regular polygon

How many sides does the polygon have?

Solution :

Number of sides of the polygon above = ?

One interior angle = 171

(n - 2)(180/n) = 171

180(n - 2) = 171n

180n - 360 = 171n

180n - 171n = 360

9n = 360

n = 360/9

n = 40

Problem 13 :

The size of each exterior angle in a regular polygon is 15°. How many sides does the regular polygon have?

Solution :

Exterior angle = 15

Interior angle = 180 - 15

= 165

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

180(n - 2) = 165n

180n - 360 = 165n

180n - 165n = 360

15n = 360

n = 360/15

n = 24

So, the required number of sides is 24.