Question

Find the magnitude of an angle which is (i) $ \dfrac{2}{3} $ of its supplement (ii) $ \dfrac{1}{4} $ of its complement.

Answer 1

Hint: To solve this problem, we need to understand the definitions of supplementary angles and complementary angles. Two angles are called supplementary when their measures add up to $ 180^\circ $ . Also, when the measures of two angles add up to $ 90^\circ $ they are called complementary. We will assume a variable for the required angles and then by using the given information, we will find the required angle.

Complete step-by-step answer:
We will first find the magnitude of an angle which is $ \dfrac{2}{3} $ of its supplement.
Let the magnitude of this angle be $ x^\circ $ . As per the definition of supplementary angles, the supplement of this angle will be $ (180 - x)^\circ $ .
I.We are given that the magnitude of the angle is $ \dfrac{2}{3} $ of its supplement.
 $
   \Rightarrow x = \dfrac{2}{3}\left( {180 - x} \right) \\
   \Rightarrow 3x = 360 - 2x \\
   \Rightarrow 3x + 2x = 360 \\
   \Rightarrow 5x = 360 \\
   \Rightarrow x = \dfrac{{360}}{5} \\
   \Rightarrow x = 72 \;
  $
Thus, our first answer is: the magnitude of the angle which is $ \dfrac{2}{3} $ of its supplement is $ 72^\circ $ .
So, the correct answer is “ $ 72^\circ $ ”.

II.Now we will find the magnitude of an angle which is $ \dfrac{1}{4} $ of its complement.
Let the magnitude of this angle be $ x^\circ $ . As per the definition of complementary angles, the complement of this angle will be $ (90 - x)^\circ $ .
We are given that the magnitude of the angle is $ \dfrac{1}{4} $ of its complement.
 $
   \Rightarrow x = \dfrac{1}{4}\left( {90 - x} \right) \\
   \Rightarrow 4x = 90 - x \\
   \Rightarrow 4x + x = 90 \\
   \Rightarrow 5x = 90 \\
   \Rightarrow x = \dfrac{{90}}{5} \\
   \Rightarrow x = 18 \;
  $
Thus, our first answer is: the magnitude of the angle which is $ \dfrac{1}{4} $ of its complement is $ 18^\circ $ .
So, the correct answer is “ $ 18^\circ $ ”.

Note: In this problem, we have formed linear equations in both the cases by with the help of definitions of supplementary and complementary angles respectively. After that, we have determined the value of the angles by solving these linear equations. Thus, in this type of question, always form the linear equations by using the given conditions and then solve it to get the final answer.