Question

Find the angle whose supplement is four times its complement.

Answer 1

Hint: To solve the question, we have to assume an angle and apply the definition of complementary angles; we will get an equation in terms of the assumed angle. Now apply the definition of supplementary angles to obtain another equation of the assumed angle. To calculate the answer, solve the above obtained equations.

Complete step-by-step answer:

Let x be the required angle.
We know that angle \[\alpha \] is called a supplement of angle \[\beta \] when the sum of the angles is equal to \[{{180}^{0}}\] .
\[\alpha +\beta ={{180}^{0}}\]
Thus, supplement of angle x is \[{{180}^{0}}-{{x}^{0}}\]
We know that angle \[\alpha \] is called a complement of angle \[\beta \] when the sum of the angles is equal to \[{{90}^{0}}\].
\[\alpha +\beta ={{90}^{0}}\]
Thus, complement of angle x is \[{{90}^{0}}-{{x}^{0}}\]
Given that supplement of an angle is four times the complement of the angle.
Thus, the supplement of angle x is four times the complement of the angle x.
By applying the obtained values in the given statement, we get
\[\begin{align}
  & {{180}^{0}}-x=4\left( {{90}^{0}}-x \right) \\
 & {{180}^{0}}-x=4\times {{90}^{0}}-4x \\
 & {{180}^{0}}-x={{360}^{0}}-4x \\
\end{align}\]
By rearranging the terms of the above equation, we get
\[{{180}^{0}}-{{360}^{0}}=x-4x\]
The signs of the terms are changed since the terms of the equation have changed their positions.
\[-{{180}^{0}}=-3x\]
By cancelling the common negative sign on both sides of the equation, we get
\[\begin{align}
  & {{180}^{0}}=3x \\
 & \Rightarrow x=\dfrac{{{180}^{0}}}{3} \\
 & x=\dfrac{3\times {{60}^{0}}}{3} \\
 & \Rightarrow x={{60}^{0}} \\
\end{align}\]
Thus, the value of angle x is equal to \[{{60}^{0}}\]
Thus, the supplement of \[{{60}^{0}}\] is four times the complement of \[{{60}^{0}}\].

Note: To solve the question, we have to apply the concepts of complementary angles and supplementary angles. To solve further, apply the given condition and solve the question obtained to calculate the required angle. Students often make mistakes with supplementary and complementary angles.