Question

Two supplementary angles differ by ${48^\circ }$ . Find the angles.

Answer 1

Hint: To solve these questions write the given information in the form of a linear equation in two variables and by applying the properties of supplementary angles. Express one angle in the terms of another angle and then simplify the equation to get the final answer.

Formula used: If two angles $x$ and $y$ are supplementary then the sum of those two angles will be ${180^\circ }$ , that is $x + y = {180^\circ }$ .

Complete step-by-step solution:
Given that two angles are supplementary. Let the angles be $x$ and $y$ . Since the angles are supplementary, we get the first equation as:
$x + y = {180^\circ }$……...…..$\left( i \right)$
Also, it is given in the question that the two angles differ by ${48^\circ }$ , this means that the larger angle is ${48^\circ }$ more than the smaller angle. Let the larger angle be $x$ and the smaller angle be $y$ . Then, according to the question, we get the equation as:
$x - y = {48^\circ }$$...............(ii)$
Adding equations $(i)$ and $(ii)$ , we get:
$2x = {228^\circ }$
Now, dividing both the sides of the equation by $2$ , we get:
$x = {114^\circ }$
This is the value of one of the angles, which is the larger of the two angles. To get the value of the smaller angle, substitute the value of $x$ in equation $(i)$ .
Substituting the value, we get:
${114^\circ } + y = {180^\circ }$
Solving the above equation for $y$ , we get:
$y = {180^\circ } - {114^\circ }$
$\Rightarrow y = {66^\circ }$

Hence, the two supplementary angles which differ by ${48^\circ }$ will be:
$x = {114^\circ }$ And $y = {66^\circ }$

Additional Information: In geometry angles including supplementary angles and many others are formed when two or more lines, usually parallel, are cut by a transversal. Some of the other angles formed by a transversal apart from supplementary angles include corresponding angles, alternate interior angles, vertically opposite angles, and alternate exterior angles.

Note: After finding one angle always remember to substitute its value in one of the equations and find the other angle. Also, you can check whether your answer is right by adding both the angles. If the sum of both the angles comes out as ${180^\circ }$ then your calculations and answer are correct.