Question

Two supplementary angles differ by \[{34^ \circ }\]. Find the angles.

Answer 1

Hint: To solve this question, we should first write the definition of supplementary angles. Supplementary angles are the two angles whose sum is \[{180^ \circ }\]. After this we will assume one of the angles as \[x\] then other becomes \[\left( {{{180}^ \circ } - x} \right)\] and then we will equate their difference to \[{34^ \circ }\] as given in the question.

Complete step by step solution:
We have given in the question the difference between two supplementary angles.
Supplementary angles are the two angles whose sum is \[{180^ \circ }\] whereas complementary angles are the two angles whose sum is \[{90^ \circ }\].
Let one angle be \[x\] and the second angle be \[y\].
Since, it is given in the question that these two angles are supplementary. Therefore, we can write \[x + y = {180^ \circ }\].
On taking \[x\] from L.H.S. to R.H.S. we get the value of \[y\] as
\[ \Rightarrow y = {180^ \circ } - x\]
Therefore, we can take the first supplementary angle as \[x\] and other as \[\left( {{{180}^ \circ } - x} \right)\].
As given in the question, these two supplementary angles differ by \[{34^ \circ }\].
Therefore, we can write
\[ \Rightarrow \left( {{{180}^ \circ } - x} \right) - x = {34^ \circ }\]
On simplifying we get
\[ \Rightarrow {180^ \circ } - 2x = {34^ \circ }\]
Taking \[{180^ \circ }\] from L.H.S. to R.H.S., we get
\[ \Rightarrow - 2x = {34^ \circ } - {180^ \circ }\]
On solving,
\[ \Rightarrow - 2x = - {146^ \circ }\]
On dividing both the sides by \[\left( { - 2} \right)\] we get
\[ \Rightarrow x = {73^ \circ }\]
\[\therefore {\text{First angle}} = {73^ \circ }\]
To find the second angle which is supplementary to this first angle, we have to subtract the first angle from \[{180^ \circ }\].
Hence, we can write
\[{\text{Second angle}} = {180^ \circ } - x\]
Putting the value of \[x\], we get
\[ \Rightarrow {\text{Second angle}} = {180^ \circ } - {73^ \circ }\]
\[ = {107^ \circ }\]
\[\therefore {\text{Second angle}} = {107^ \circ }\]
So, the correct answer is “\[{73^ \circ }\] and \[{107^ \circ }\]”.

Note: To solve this problem, the most important thing is the definition of supplementary angles. In place of supplementary angles if complementary angle is given then we have to take angles as \[x\] and \[\left( {{{90}^ \circ } - x} \right)\] because complementary angles are two angles whose sum is \[{90^ \circ }\].