Question

The measure of the supplement of an angle is ${{20}^{\circ }}$ more than three times the measure of the original angle. What are the measures of the angles?

Answer 1

Hint: We have to find the measure of the two angles. Firstly by using our statement we will form the value of two angles in an unknown variable form. Then as we know that the sum of supplementary angles is ${{180}^{\circ }}$ we will write the sum of the two angles equal to it. Finally we will simplify the equation and get our desired answer.

Complete step by step solution:
We have to find the two angles.
Firstly we have been given that supplement of an angle is ${{20}^{\circ }}$ more than three times the measure of the original angle
Let the original angle be $x$ then according to the above statement the value of the supplementary angle will be as follows:
$3x+{{20}^{\circ }}$……$\left( 1 \right)$
Now we know that the sum of supplementary angles is ${{180}^{\circ }}$. So,
$x+\left( 3x+{{20}^{\circ }} \right)={{180}^{\circ }}$
$\Rightarrow 4x+{{20}^{\circ }}={{180}^{\circ }}$
Simplifying further we get,
$\Rightarrow 4x={{180}^{\circ }}-{{20}^{\circ }}$
$\Rightarrow 4x={{160}^{\circ }}$
Dividing both sides by $4$ we get,
$\Rightarrow \dfrac{4x}{4}=\dfrac{{{160}^{\circ }}}{4}$
$\Rightarrow x={{40}^{\circ }}$
So we get the original angle which is $x$ as ${{40}^{\circ }}$
Substitute $x={{40}^{\circ }}$ in equation (1) we get,
$\Rightarrow 3\times {{40}^{\circ }}+{{20}^{\circ }}$
$\Rightarrow {{120}^{\circ }}+{{20}^{\circ }}$
So,
$\Rightarrow {{140}^{\circ }}$
So we get the answer as ${{40}^{\circ }},{{140}^{\circ }}$ .
Hence the measures of two angles are ${{40}^{\circ }},{{140}^{\circ }}$.

Note:
In Geometry, study of lines and angles is basic and important to understand the advanced topics.
> Two angles are said to be supplementary if their sum is ${{180}^{\circ }}$ which means the two angles together make a straight line but the angle need not be together.
> Two angles are said to be complementary if their sum is ${{90}^{\circ }}$ which means the two angles together make perpendicular lines.