Question

The ratio of the measure of an exterior angle of a regular 7:2 nonagon to the measure of one of its interior angles is:
(a) 7:2
(b) 2:7
(c) 4:3
(d) 3:4

Answer 1

Hint:To solve the given question, first we will find what the total number of sides in a regular nonagon is. Then we will find the sum of the interior angles of the nonagon by applying the formula: Sum of interior angles \[=\left( n-2 \right)\times {{180}^{\circ}}.\] Then we will divide the sum of the interior angles by the number of sides to get the measure of each interior angle. Then we will subtract this interior angle from \[{{180}^{\circ}}\] to get the measure of the exterior angle. Then, finally, we will take the ratio of both.

Complete step by step answer:
Before we solve this question, we must know that the number of sides in a nonagon is 9. Now, we will find the sum of the interior angles in a regular nonagon. The formula for calculating the sum is given by
\[\text{Sum of interior angles}=\left( n-2 \right)\times {{180}^{\circ}}\]
where n is the number of sides in the polygon. In our case, n = 9. Thus, we will get,
\[\Rightarrow \text{Sum of interior angles}=\left( 9-2 \right)\times {{180}^{\circ}}\]
\[\Rightarrow \text{Sum of interior angles}=7\times {{180}^{\circ}}\]
\[\Rightarrow \text{Sum of interior angles}={{1260}^{\circ}}\]
Now, we have to find the measure of each interior angle. This will be obtained by dividing the sum of the interior angles by the number of sides. Thus, we will get,
\[\text{Interior angle in a regular nonagon}=\dfrac{{{1260}^{\circ}}}{9}={{140}^{\circ}}\]

Thus, the interior angle in a regular nonagon is \[{{140}^{\circ}}.\] Now, we will find the exterior angle of this regular nonagon. We know that in any polygon, the sum of the interior and exterior angles is \[{{180}^{\circ}}.\] Thus, we have,
\[\text{Interior angle + Exterior angle}={{180}^{\circ}}\]
\[\Rightarrow {{140}^{\circ}}\text{+ Exterior angle}={{180}^{\circ}}\]
\[\Rightarrow \text{Exterior angle}={{180}^{\circ}}-{{140}^{\circ}}\]
\[\Rightarrow \text{Exterior angle}={{40}^{\circ}}\]
Now, we will take their ratio. Thus, we have,
\[\text{Ratio}=\dfrac{\text{Exterior Angle}}{\text{Interior Angle }}\]
\[\Rightarrow \text{Ratio}=\dfrac{\text{4}{{\text{0}}^{\circ}}}{\text{14}{{\text{0}}^{\circ}}\text{ }}\]
\[\Rightarrow \text{Ratio}=\dfrac{2}{7}\]
Hence, option (b) is the right answer.

Note:
The alternate method of solving the question is given below. The exterior angle of the polygon is \[\left( \dfrac{{{360}^{\circ}}}{n} \right)\] where n is the number of sides in the polygon. In our case, n = 9. Thus, the exterior angle will be \[\left( \dfrac{{{360}^{\circ}}}{9} \right)={{40}^{\circ}}.\] Also,
\[\text{Interior angle + Exterior angle}={{180}^{\circ}}\]
 \[\Rightarrow \text{Interior angle }+{{40}^{0}}={{180}^{\circ}}\]
\[\Rightarrow \text{Interior angle}={{140}^{\circ}}\]
Thus, the ratio becomes
\[\text{Ratio}=\dfrac{\text{4}{{\text{0}}^{\circ}}}{\text{14}{{\text{0}}^{\circ}}\text{ }}=\dfrac{2}{7}\]