Question

One angle of a decagon is 90˚ and all the remaining nine angles are equal. What is the measure of the other angles?

Answer 1

Hint: In order to find the solution of the given question, we need to remember that the sum of all interior angles of a ‘n’ sided polygon is given by the formula, (n - 2) (180˚). So, if n = 3, that is for a triangle we get, (3 - 2) (180˚) = 180˚ as the sum of all interior angles. So, by using this concept, we can find the solution to this question.

Complete step-by-step answer:
In this question, we should know that the sum of all interior angles of an ‘n’ sided polygon is given by the formula, (n - 2) (180˚). So, we can say that for a 10 sided polygon, that is for a decagon, we get, (10 - 2) (180˚) = 1440˚ as the sum of all interior angles.
Now, according to the question, we have been given that one angle is 90˚ and all the other 9 angles are equal. So, let us consider the measure of each angle as x. So, we can write the given condition as,
90˚ + 9 (x) = 1440˚
Now, we will simplify it further to get the value of x. So, we can write,
9 x = 1440˚ - 90˚
9 x = 1350˚
Now, we will divide the whole equation by 9. So, we get,
$\begin{align}
  & \dfrac{9x}{9}=\dfrac{{{1350}^{\circ }}}{9} \\
 & x={{150}^{\circ }} \\
\end{align}$
Hence, we get the measure of each equal angle as 150˚.

Note: While solving this question, there are possibilities of calculation mistakes. So, we have to be very careful while doing the calculations. We can also solve this question by taking the average of all the angles equal to one angle of a regular decagon and then simplifying to get the value of x.