Question

How many obtuse angles in a regular pentagon?

Answer 1

Hint: First we will specify the number of angles in pentagon and its properties. Then we will evaluate the value of one angle of a pentagon and similarly the value of all the other remaining values of pentagon. Then define obtuse angle and its properties.

Complete step-by-step answer:
We will start by evaluating the summation of the interior angles of the pentagon. We can evaluate the sum of the interior angles in a pentagon by the following formula:
 $ {180^0}(n - 2) $ , where $ n $ is the number of the sides of a polygon.
As, here it is pentagon, hence the value of $ n $ will be $ 5 $ .
 $
   = {180^0}(n - 2) \\
   = {180^0}(5 - 2) \\
   = {180^0}(3) \\
   = {540^0} \;
  $
Now, here as the pentagon is a regular polygon, this means that all of the $ 5 $ angles are equal to one another. We can evaluate the degrees of one interior angle by doing the following:
 $
   = \dfrac{{540}}{5} \\
   = {108^0} \;
  $
Since, an obtuse angle is greater than $ {90^0} $ but smaller than $ {180^0} $ . So, this means that $ {108^0} $ must be an obtuse angle. Since, here there are a total five $ {108^0} $ angles in the pentagon, so we can say that there are five obtuse angles in a regular pentagon.
Hence, there are total $ 5 $ obtuse angles in a regular pentagon.
So, the correct answer is “ $ 5 $ ”.

Note: Remember that an obtuse angle is an angle which is greater than $ {90^0} $ but smaller than $ {180^0} $ and in an acute angle, the acute angle is smaller than $ {90^0} $ . Also, while evaluating the value of an angle of a polygon be careful with the calculations. Make sure to substitute the values properly.