Question

Among the two supplementary angles, the measure of the larger angle is \[{36^ \circ }\] more than the measure of the smaller. Find their measures.

Answer 1

Hint: We assume the smaller angle as a variable and using the relation in the question we write the larger angle in terms of smaller angle. Using the concept of supplementary angles we add both the larger and smaller angles and equate the sum to ${180^ \circ }$
* Supplementary angles are the set of angles which lie on a straight line.

Complete step-by-step answer:
Let the smaller angle \[ = x\]
The larger angle \[ = y\]
We find the relation between both supplementary angles
Since the larger angle is more than the smaller angle by \[{36^ \circ }\] therefore we add the value to the smaller angle and equate it to the larger angle.
$y = x + 36$ ... (1)
Since both the angles are supplementary, then their sum has to be ${180^ \circ }$
$x + y = {180^ \circ }$
Substitute the value of $y = x + 36$ from equation (1)
$x + (x + 36) = {180^ \circ }$
$2x + 36 = {180^ \circ }$
Shift all constant terms in degrees to one side of the equation.
$
  2x = {180^ \circ } - {36^ \circ } \\
  2x = {144^ \circ } \\
 $
Dividing both sides by 2
\[\dfrac{{2x}}{2} = \dfrac{{{{144}^ \circ }}}{2}\]
\[x = {72^ \circ }\] … (2)
Substitute the value of \[x = {72^ \circ }\]in equation (1) and solve for the value of the larger angle.
$y = {72^ \circ } + {36^ \circ }$
$y = {108^ \circ }$
This, Measure of the smaller angle, $x = {72^ \circ }$
And measure of the greater angle, $y = {108^ \circ }$

Note: Students can many times make mistake of forming the equation with more than part wrong because they think that value is added to the part which is given more but students should keep in mind that the value is always added to the lesser part and equate it to the part which is given more. Students should always write their answer along with the degree sign and convert all the values in the same unit. Conversion for angle is \[{1^ \circ } = \dfrac{\pi }{{{{180}^ \circ }}}\].