Question

The angle which is one fifth of its supplement is
(a) \[15^\circ \]
(b) \[30^\circ \]
(c) \[45^\circ \]
(d) \[60^\circ \]

Answer 1

Hint: Here, we need to find the angle which is one fifth of its supplement. We will assume the supplement of the required angle be of measure \[x\] degrees. We will use the property of supplementary angles to form an equation in terms of \[x\]. We will solve this equation to find the value of \[x\], and hence, the measure of the supplement angle and the required angle.

Complete step-by-step answer:
Let the supplement of the required angle be of measure \[x\] degrees.
It is given that the required angle is one fifth of its supplement angle.
Thus, we get
Required angle \[ = \dfrac{1}{5}{\rm{ of }}x\]
Simplifying the expression, we get
Required angle \[ = \dfrac{x}{5}\]
Now, we know that any two angles are called supplements of each other if their sum is equal to 180 degrees.
The required angle and its supplement are supplementary angles.
Therefore, we get the linear equation
\[ \Rightarrow \dfrac{x}{5} + x = 180^\circ \]
This is a linear equation in one variable in terms of \[x\]. We will solve the equation further to find the value of \[x\].
Rewriting the equation, we get
\[ \Rightarrow \dfrac{x}{5} + \dfrac{x}{1} = 180^\circ \]
The L.C.M. of 5 and 1 is 5.
Rewriting the expression with denominator 5, we get
\[ \Rightarrow \dfrac{x}{5} + \dfrac{{5x}}{5} = 180^\circ \]
Adding the terms of the expression, we get
\[\begin{array}{l} \Rightarrow \dfrac{{x + 5x}}{5} = 180^\circ \\ \Rightarrow \dfrac{{6x}}{5} = 180^\circ \end{array}\]
Multiplying both sides by 5, we get
\[ \Rightarrow 6x = 900^\circ \]
Dividing both sides by 6, we get
\[ \Rightarrow x = 150^\circ \]
Therefore, we get the measure of the supplement of the required angle as \[150^\circ \].
Substituting \[x = 150^\circ \] in the expression \[\dfrac{x}{5}\], we get
\[ \Rightarrow \] Required angle \[ = \dfrac{{150^\circ }}{5}\]
Simplifying the expression, we get
\[ \Rightarrow \] Required angle \[ = 30^\circ \]
Therefore, we get the measure of the required angle as \[30^\circ \].
Thus, the correct option is option (b).

Note: We have formed a linear equation in one variable in terms of \[x\] in the solution. A linear equation in one variable is an equation that can be written in the form \[ax + b = 0\], where \[a\] is not equal to 0, and \[a\] and \[b\] are real numbers. For example, \[x - 100 = 0\] and \[100P - 566 = 0\] are linear equations in one variable \[x\] and \[P\] respectively. A linear equation in one variable has only one solution or we can say it has only one root.