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Two supplementary angles are in the ratio \[5:4\]. Find the angles.

Last updated date: 20th Jun 2024
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Hint: Here we assume both the angles as separate variables. Then using the information in the question we form an equation of sum of angles. Also, using the concept of ratio, we write the angles in the ratio form and equate it to the given ratio.
* Two angles are said to be supplementary to each other if they have the sum of angles as \[{180^ \circ }\]
* Ratio \[m:n\]can be written in the form of fraction as \[\dfrac{m}{n}\].

Complete answer:
Let us assume one angle as ‘x’ and another angle as ‘y’.
Since we know both the angles are supplementary angles, then their sum must be equal to \[{180^ \circ }\]
So we can write the sum of angles x and y as \[{180^ \circ }\]
\[ \Rightarrow x + y = {180^ \circ }\] … (1)
Now we are given that the angles are in the ratio \[5:4\]
So we can write that the ratio of angles x and y is \[5:4\].
Substitute the values of angles we assumed in the beginning of the solution.
\[ \Rightarrow x:y = 5:4\] … (2)
Now since we know we can convert the ratio \[a:b\] into a form of fraction as \[\dfrac{a}{b}\].
Therefore, we can write the equation (2) as
\[ \Rightarrow \dfrac{x}{y} = \dfrac{5}{4}\]
Multiply both sides of the equation by y
\[ \Rightarrow \dfrac{x}{y} \times y = \dfrac{5}{4} \times y\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow x = \dfrac{5}{4}y\] … (3)
Substitute the value of x from equation (3) in equation (1).
\[ \Rightarrow \dfrac{{5y}}{4} + y = {180^ \circ }\]
Take LCM on the left hand side of the equation.
\[ \Rightarrow \dfrac{{5y + 4y}}{4} = {180^ \circ }\]
Calculate the sum in the numerator.
\[ \Rightarrow \dfrac{{9y}}{4} = {180^ \circ }\]
Multiply both sides by \[\dfrac{4}{9}\]
\[ \Rightarrow \dfrac{{9y}}{4} \times \dfrac{4}{9} = {180^ \circ } \times \dfrac{4}{9}\]
Cancel out the same terms from numerator and denominator.
\[ \Rightarrow y = {\left( {20 \times 4} \right)^ \circ }\]
Calculate the product.
\[ \Rightarrow y = {80^ \circ }\]
Now we substitute the value of y in equation (1) to calculate the value of x.
\[ \Rightarrow x + {80^ \circ } = {180^ \circ }\]
Shift all constants to one side of the equation.
\[ \Rightarrow x = {180^ \circ } - {80^ \circ }\]
Calculate the value on RHS.
\[ \Rightarrow x = {100^ \circ }\]

\[\therefore \]Two supplementary angles that are in the ratio \[5:4\] are \[{100^ \circ }\] and \[{80^ \circ }\]

Students many times make the equation formed by the ratio as a complex equation when they cross multiply the values to both sides and then solve. Always try to keep that value on one side of the equation which can later be directly substituted in another equation. Also, keep in mind ratio should always be in simplest form i.e. there should not be any common factor between numerator and denominator.