Question

The angles of a convex pentagon are in the ratio $2:3:5:9:11$. Find the measure of each angle.

Answer 1

Hint: We know that the sum of the angles of a convex polygon is $\left( {n - 2} \right) \times 180^\circ $where n is the number of sides. As the ratio of the angles is given, we can add the ratio with a variable and equate their sum to the sum of the interior angles of the convex polygon. Then we can solve for the variable and find all the angles.

Complete step by step answer:

We are given that the angles of pentagon are in the ratio $2:3:5:9:11$. So, we can write the angles as $2x,{\text{ }}3x,{\text{ }}5x,{\text{ }}9x,{\text{ }}11x$ where x is a variable.
The sum of the interior angles of a convex 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 = 3 \times 180^\circ = 540^\circ $
So, for the given pentagon, the sum of the angles can be written as,
$2x + 3x + 5x + 9x + 11x = {\text{ }}540^\circ $
$ \Rightarrow 30x = {540^o}$
$ \Rightarrow x = \dfrac{{540^\circ }}{{30}} = 18^\circ $
Putting the values of x in the angles, we get,
$2x = 2 \times 18^\circ = 36^\circ $
$3x = 3 \times 18^\circ = 54^\circ $
$5x = 5 \times 18^\circ = 90^\circ $
$9x = 9 \times 18^\circ = 162^\circ $
$11x = 11 \times 18^\circ = 198^\circ $
Therefore, the angles are, $36^\circ ,{\text{ 54}}^\circ ,{\text{ 9}}0^\circ ,{\text{ 162}}^\circ ,{\text{ 198}}^\circ $

Note: Convex polygon are the polygons having all of its interior angle less than ${\text{$180^\circ$ }}$. The sum of the interior angles of a convex polygon having n sides is $\left( {n - 2} \right) \times 180^\circ $. A ratio gives us the information that how a quantity is compared to another quantity. A ratio can be multiplied of divided by same 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 them with the sum we obtained using the angle sum rule.