Question

An angle is \[\dfrac{2}{3}\] times of its supplementary. What is its magnitude?

Answer 1

Hint: An angle is given as the \[\dfrac{2}{3}\] of its supplement, Therefore here we will use the formula for supplementary angles. We will use the sum of the supplementary angles to solve this problem

Complete step-by-step answer:
Let the measure of the angle be \[x\]. And the measure of another angle be \[y\]
Since we know that, if angles are supplementary then the sum of angles would be \[180^\circ \].
That means that, sum of the angles \[x\] and \[y\] is equal \[180^\circ \]
This can be written as, \[x{\text{ }} + {\text{ }}y{\text{ }} = {\text{ }}180^\circ \]
Also it is given in the question that x is \[\dfrac{2}{3}\] of its supplementary
That is angle x is two third of the angle y
\[x = \dfrac{{2y}}{3}\]
this means that \[y = \dfrac{{3x}}{2}\]…. … …..(i)
Now, we know that \[x{\text{ }} + {\text{ }}y{\text{ }} = {\text{ }}180^\circ \]
By putting the value of y from equation (i), we will get:
\[\Rightarrow x + \dfrac{{3x}}{2} = 180^\circ \](using equation (i))
\[\Rightarrow \dfrac{{5x}}{2} = 180^\circ \]
\[\Rightarrow x = 72^\circ \]
Now we have found the value of x, we can find the value of y using value of x as:
\[\Rightarrow x{\text{ }} + {\text{ }}y{\text{ }} = {\text{ }}180^\circ \]
\[\Rightarrow y = 180^\circ - x\]
\[\Rightarrow y = 180^\circ - 72^\circ \]
\[\Rightarrow y = 108^\circ \]

Note: In such a problem we should remember that the sum of the supplementary angles is \[180^\circ \] and we should start solving to find the individual angles according to the condition in the question.