Question

The sides of a pentagon are produced in order and the exterior angle so obtained are the measures of x degree, 2x degree, (3x + 10) degree, (4x + 5) degree and 5x degree respectively. Find the value of x and the measure of each exterior angle of the polygon.

Answer 1

Hint: We know that pentagon is a polygon with 5 sides. And for any polygon the sum of exterior angles is 360 degree. Hence we have x + 2x + 3x + 10 + 4x + 5 + 5x = 360.

Complete step by step answer:
Now we are given that the sides of a pentagon are produced in order and the exterior angle so obtained are the measures of x degree, 2x degree, (3x + 10) degree, (4x + 5) degree and 5x degree respectively.
We know that pentagon is nothing but a 5 polygon with 5 sides and for any n sided polygon we have the sum of exterior angles is 360 degree.
Hence we can say that.
x + 2x + 3x + 10 + 4x + 5 + 5x = 360.
Now rearranging the terms we get.
x + 2x + 3x + 4x + 5x + 10 + 5 = 360.
Now let us take x common
(1 + 2 + 3 + 4 + 5)x + 15 = 360
15x + 15 = 360.
Now taking 15 to RHS we get
15x = 360 – 15 = 345.
Hence we have 15x = 345.
Now dividing by 15 on both sides we get
$x=\dfrac{345}{15}=23$
Hence x = 23 degrees.
Now let us substitute x in each angle to find all exterior angles
x degree = 23 degree.
2x degree = 2 × 23 = 46 degree.
(3x + 10) degree = 3 × 23 + 10 = 69 + 10 = 79 degree.
(4x + 5) degree = 4 × 23 + 5 = 92 + 5 = 97 degree.
5x degree = 5 × 23 = 115 degree.

Hence the exterior angles of polygons are 23 degree, 46 degree, 79 degree, 97 degree, 115 degree.

Note: Note that exterior angles and interior angles are different. Sum of all exterior angles of pentagon is 360 degrees while the sum of all interior angles of pentagon is 540 degrees. In general for any n sided polygon we have sum of exterior angles is 360 degree and sum of interior angles is (n – 2) × 180 degree.