Question

The angles pentagon in degrees are x, x + 20, x + 40, x + 60, x + 80, then the smallest angle of the pentagon is
(A). $$50^{\circ}$$
(B). $$68^{\circ}$$
(C). $$78^{\circ}$$
(D). $$85^{\circ}$$

Answer 1

Hint: In this question it is given that the angles pentagon in degrees are x, x + 20, x + 40, x + 60, x + 80, then we have to find the smallest angle of the pentagon i.e, the value of x. So to find the solution we need to know that the sum of all angles of a pentagon is $$540^{\circ}$$. So by using this information we have to find the solution.

Complete step-by-step solution:
Given the angles(in degre) of the pentagon are x, x + 20, x + 40, x + 60 and x + 80.
As we know that sum of angle of pentagon = $$540^{\circ}$$
Therefore,
$$x+\left( x+20^{\circ}\right) +\left( x+40^{\circ}\right) +\left( x+60^{\circ}\right) +\left( x+80^{\circ}\right) =540^{\circ}$$
$$\Rightarrow x+x+20^{\circ}+x+40^{\circ}+x+60^{\circ}+x+80^{\circ}=540^{\circ}$$
$$\Rightarrow 5x+200^{\circ}=540^{\circ}$$
$$\Rightarrow 5x=540^{\circ}-200^{\circ}$$
$$\Rightarrow 5x=340^{\circ}$$
$$\Rightarrow x=\dfrac{340^{\circ}}{5}$$
$$\Rightarrow x=68^{\circ}$$
Since, among all the angles the smallest angle is x then our measure of the smallest angle is $$68^{\circ}$$.
Hence the correct option is option B.

Note: You may got different types of polygon related problem where you need to know about the sum of the angles, so we can find it by using a particular formula which states that, if n be the number of sides of a polygon then the sum of the all angle of a polygon is $$\left( n-2\right) \times 180^{\circ}$$.
So for the above problem we can also find the sum of the alls angles of the given pentagon,
Since it is a pentagon, so n=5,
Therefore, by the above formula we can write,
$$\left( n-2\right) \times 180^{\circ}$$
$$=\left( 5-2\right) \times 180^{\circ}$$
$$=3\times 180^{\circ}$$
$$=540^{\circ}$$
So while solving if you don’t know the sum of the angles then you can use the above method in order to find the summation.