Question

Two angles are supplementary. The larger angle measures $120$ degrees more than the smaller. What is the degree measure of each angle?

Answer 1

Hint: Here we are asked to find the measures of the angles that are supplementary by using the given data. For that, we will first make the given statement into an expression for each measure then it is given that the angles are supplementary, that is the sum of those angles equal $180$ degrees. So, by equating the sum to $180$ we can find the measure of each angle.


Complete step by step answer:
It is given that the angles are supplementary and the largest angle is one hundred and twenty degrees more than the smaller angle. We aim to find the measure of each angle.
Let $\angle A$ and $\angle B$ be the two angles and let $\angle A$ be the largest angle and $\angle B$ be the smaller angle. It is given that the larger angle is one hundred and twenty degrees more than the smaller angle.
So, $\angle A = \angle B + 120^\circ $
Now the two angles are $\angle B + 120^\circ $ and $\angle A$ .
Since it is given that the angles are supplementary angles their sum will be equal to one hundred and eighty degrees.
So, we can write $\left( {\angle B + 120} \right) + \angle B = 180$
Now let us simplify the above to find the measure of $\angle B$
$\left( {\angle B + 120} \right) + \angle B = 180 \Rightarrow \angle B + 120 + \angle B = 180$
$ \Rightarrow 2\angle B + 120 = 180$
Taking one hundred and twenty to the other side we get
$ \Rightarrow 2\angle B = 180 - 120$
On simplifying the above, we get
$ \Rightarrow 2\angle B = 60$
On further simplification we get
$ \Rightarrow \angle B = \dfrac{{60}}{2}$
$ \Rightarrow \angle B = 30$
Thus, we have found that the measure of the angle $\angle B$ is $30^\circ $ .
Now let us substitute the measure of an angle $\angle B$ in angle $\angle A$ to find its measure.
$\angle A = \angle B + 120 \Rightarrow \angle A = 30 + 120$
$\angle A = 150$
Now we also got the measure of another angle $\angle A$ as $150^\circ $ .
Hence, the measure of the two angles is $150^\circ $ and $30^\circ $ .

Note: When a ray cuts a line which makes the line into two then there will be two angles then these angles are called adjacent angles. When the sum of two angles totals ninety degrees then those angles are called complementary angles. That is if $\angle P$ and $\angle Q$ are the complementary angles then $\angle P + \angle Q = 90^\circ $ .