Question

The interior angles of a pentagon are in the ratio $4:5:6:7:5$. Find each angles of the pentagon
A.$80^\circ ,100^\circ ,120^\circ ,140^\circ {\text{and }}100^\circ $
B.$100^\circ ,120^\circ ,110^\circ ,140^\circ {\text{and }}100^\circ $
C.\[60^\circ ,100^\circ ,120^\circ ,130^\circ {\text{and }}110^\circ \]
D.None of the above

Answer 1

Hint: We know that the sum of the angles of a polygon is $\left( {n - 2} \right) \times 180^\circ $where n is the number of sides. Here we have \[n = 5\] as we have a pentagon. As the ratio of the angles are given, we can add the ratio with a variable and equate their sum to the sum of the interior angles of the pentagon. Then we can solve for the variable and find all the angles by substituting back in the ratio.

Complete step-by-step answer:

We are given that the angles of the pentagon are in the ratio $4:5:6:7:5$. So, we can write the angles as $4x,{\text{ 5}}x,{\text{ 6}}x,{\text{ 7}}x,{\text{ 5}}x$ where x is a variable.
The sum of the interior angles of a polygon is given by the equation,
$S = \left( {n - 2} \right) \times 180^\circ $ where n is the number of sides.
For a pentagon, n becomes 5 and the sum of the angle becomes,
$S = \left( {5 - 2} \right) \times 180^\circ $
On simplifying the bracket we get,
$S = 3 \times 180^\circ $
On multiplication we get,
$S = 540^\circ $
So, for the given pentagon, the sum of the angles can be written as,
$4x + 5x + 6x + 7x + 5x = {\text{ }}540^\circ $
$ \Rightarrow 27x = {540^o}$
On dividing the equation by 27 we get,
$ \Rightarrow x = \dfrac{{540^\circ }}{{27}} = 20^\circ $
Putting the values of $x = 20^\circ $ in the angles, we get,
$ \Rightarrow 4x = 4 \times 20 = 80^\circ $
$ \Rightarrow 5x = 5 \times 20 = 100^\circ $
$ \Rightarrow 6x = 6 \times 20 = 120^\circ $
$ \Rightarrow 7x = 7 \times 20 = 140^\circ $
$ \Rightarrow 5x = 5 \times 20 = 100^\circ $
Therefore, the angles are, $80^\circ ,100^\circ ,120^\circ ,140^\circ ,100^\circ $.
So, the correct answer is option A.

Note: A ratio gives us the information that how a quantity is compared to another quantity. A ratio can be multiplied or divided by the same value to get the required quantity. In this question, we take x as the value that to be multiplied. Then we solved for x and multiplied it with the ratio to get the required angles. As this problem includes a lot of multiplications, there are chances of making mistakes. We must multiply the ratio with a variable before applying the sum of the interior angles. We can verify our answers by taking the sum of the angles and comparing with the sum we obtained using the angle sum rule.
An alternate approach to the problem is to compare the option. As the question comes with options, we can compare them with ratio. As ratio of \[{2^{nd}}\] and \[{5^{th}}\] angles are equal, the option with \[{2^{nd}}\]and \[{5^{th}}\] angles equal will be the correct answer. We can also check the sum of the option and the option having sum of ${540^o}$will be the answer.