Question

The larger of two supplementary angles exceeds $7$ times the smaller by $4$ degrees. What is the measure of the larger angle?

Answer 1

Hint: Two angles are said to be supplementary angles if they add up to $180$ degrees. Supplementary angles form a straight angle $(180^\circ )$ when they are put together. ”S” in supplementary stands for straight lines as they form $180^\circ $. In other words, $\angle 1\,$ and $\angle 2$ are said to be supplementary, if $\angle 1 + \angle 2 = 180^\circ $

Complete step by step solution:
Let us assume
Larger angle $ = L$
Smaller angle $ = S$
It is given that these two angles are supplementary angles, it means
$L + S = 180^\circ \ldots \ldots (1)$
It is given that $L$ exceeds $7$ times of $S$ by $4^\circ $
So,
$L = (7 \times S) + 4 = 7S + 4$
Now put the above value of $L$ in equation $(1)$ we get
$(7S + 4) + S = 180^\circ $
Now adding the $S$ terms. We get,
$8S + 4 = 180^\circ $
Shifting $4$ to the right side of the equation. We get,
$ \Rightarrow 8S = 180 - 4$
$ \Rightarrow $$8S = 176$.
$ \Rightarrow S = \dfrac{{176}}{8}$
$ \Rightarrow S = 22^\circ $
Now we can find the value of $L$ by putting the value $S = 22^\circ $ in equation $(1)$. We get,
$ \Rightarrow L + 22^\circ = 180^\circ $
Shifting $22$ to the right side of the equation. We get,
$ \Rightarrow L = 180^\circ - 22^\circ $
$ \Rightarrow L = 158^\circ $
Hence, the measure of Smaller angle$ = 22^\circ $and the measure of the larger angle is $158^\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 a 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 pairs. The definition of supplementary angles holds true only for two angles.