Question

The ratio of two supplementary angles is $2:7$. What is the measure of the smaller angle?

Answer 1

Hint: Here we have two supplementary in the ratio $2:7$ and we have to find the measure of the smaller angle. Two angles are said to be supplementary angles if they add up to $180$ degrees. Supplementary angles form a straight angle $(180\deg )$ when they are put together. ”S” in supplementary stands for straight lines as they form $180^\circ $.

Complete step-by-step solution:
Supplementary angles can be defined as the angles when the sum of the measure of the two angles is $180^\circ $, then the pair is said to be supplementary angles. It is not necessary that the supplementary angles are always adjacent to each other; they can be different; only their sum should be $180^\circ $.
In other words, $\angle 1\,$and $\angle 2$ are said to be supplementary, if $\angle 1 + \angle 2 = 180^\circ $
Here we have two supplements in the ratio $2:7$.
So, let us assume the two angles be $2x$ and $7x$
It is given that these two angles are supplementary angles, it means
$ \Rightarrow 2x + 7x = 180^\circ $
Adding $x$ terms we get,
$ \Rightarrow 9x = 180^\circ $
On dividing $180^\circ $ by $9$ we get,
$ \Rightarrow x = \dfrac{{180^\circ }}{9} = 20^\circ $
So, $x = 20^\circ $
Therefore, the two angles will be
$ \Rightarrow 2 \times 20^\circ = 40^\circ $
$ \Rightarrow 7 \times 20^\circ = 140^\circ $

Hence the measure of the smaller angle is $20^\circ $.

Note: When the sum of two pairs of angles is equal to $180^\circ $, then we call that pair of angles, supplements of each other. So, we know that the sum of two supplementary angles is $180^\circ $, and each of them is said to be supplement of each other. Thus, the supplement of an angle is found by subtracting it from $180^\circ $. This means the supplement of $x^\circ $ is $(180 - x)^\circ $. Note that three angles can never be supplementary even though their sum is $180^\circ $because supplementary angles always occur in pair. The definition of supplementary angles holds true only for two angles.