Question

$ ABCDE $ is the regular pentagon, diagonal $ AD $ divides $ \angle CDE $ in two parts, what is the ratio of $ \angle ADE $ and $ \angle ADC $ ?

Answer 1

Hint: To solve this problem, we need to find the value of an interior angle of a regular pentagon. For this, we will use the formula which relates the number of sides to the interior angle of a pentagon. After that we will find the values of two angles formed by the diagonal dividing one interior angle as given in the problem.
Formula used:
 $ (n - 2) \times 180 = n \times \theta $ , where, $ n $ is the number of sides and $ \theta $ is the interior angle of regular pentagon

Complete step-by-step answer:
First, we will find the value of an interior angle of a regular pentagon.
We know that $ (n - 2) \times 180 = n \times \theta $
For a regular pentagon, the number of sides $ n $ is $ 5 $ .
 $
   \Rightarrow (5 - 2) \times 180 = 5 \times \theta \\
   \Rightarrow \theta = \dfrac{{3 \times 180}}{5} \\
   \Rightarrow \theta = {108^0} \;
  $

Now, the diagonal $ AD $ is parallel to the side $ BC $ of the pentagon.
Therefore, we can say that
 $
  \angle ADC + \angle DCB = 180 \\
   \Rightarrow \angle ADC = 180 - \angle DCB \\
   \Rightarrow \angle ADC = 180 - 108 \\
   \Rightarrow \angle ADC = {72^0} \;
  $
We know that $ \angle CDE $ is an interior angle of the regular pentagon, therefore, $ \angle CDE = {108^0} $
It is given that diagonal $ AD $ divides $ \angle CDE $ in two parts, what is the ratio of $ \angle ADE $ and $ \angle ADC $ in regular pentagon $ ABCDE $ . Therefore,
 $
  \angle ADE + \angle ADC = \angle CDE \\
   \Rightarrow \angle ADE = \angle CDE - \angle ADC \\
   \Rightarrow \angle ADE = 108 - 72 \\
   \Rightarrow \angle ADE = {36^0} \;
  $
We are asked to find the ratio of $ \angle ADE $ and $ \angle ADC $
 $ \dfrac{{\angle ADE}}{{\angle ADC}} = \dfrac{{36}}{{72}} = \dfrac{1}{2} $
Hence, the ratio of $ \angle ADE:\angle ADC = 1:2 $
So, the correct answer is “1:2”.

Note: There are three main steps to solve this question. First we have determined the interior angle of the regular pentagon by using the formula $ (n - 2) \times 180 = n \times \theta $ . After that, we have determined one of the two angles formed by the given diagonal which divides an interior angle. Then, using the value of this angle and the interior angle, we have determined the other angle. Finally, we have obtained the required ratio.