Question

Two angles are supplementary, and one is \[{5^ \circ }\] more than six times the other. What is the larger angle?

Answer 1

Hint: To solve this question, we should first know 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 using the concept of supplementary angle, the other will become \[\left( {{{180}^ \circ } - x} \right)\]. But, according to the question, one angle is \[{5^ \circ }\] more than six times the other angle. Therefore, the other angle will become \[\left( {6x + {5^ \circ }} \right)\] and then we will equate \[\left( {{{180}^ \circ } - x} \right)\] with \[\left( {6x + {5^ \circ }} \right)\] to find the value of \[x\] and then the larger angle.

Complete step by step answer:
We have given in the question that there are two supplementary angles and one is \[{5^ \circ }\] more than six times the other.
Supplementary angles are the two angles whose sum is \[{180^ \circ }\] whereas complementary angles are the two angles whose sum is \[{90^ \circ }\].
Let the smaller angle be \[x\] and the larger 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
\[ \Rightarrow y = {180^ \circ } - x - - - (1)\]
Also, given in the question that one angle is \[{5^ \circ }\] more than six times the other angle.
We can write,
\[ \Rightarrow y = 6x + {5^ \circ } - - - (2)\]
On equating \[(1)\] and \[(2)\], we get
\[ \Rightarrow {180^ \circ } - x = 6x + {5^ \circ }\]
Adding \[x\] on both the sides of the equation, we get
\[ \Rightarrow {180^ \circ } = 6x + {5^ \circ } + x\]
Subtracting \[{5^ \circ }\] from both the side of the equation, we get
\[ \Rightarrow {180^ \circ } - {5^ \circ } = 6x + x\]
On simplification, we get
\[ \Rightarrow {175^ \circ } = 7x\]
On rearranging we get
\[ \Rightarrow 7x = {175^ \circ }\]
Dividing both the sides by \[7\], we get
\[ \Rightarrow x = {25^ \circ }\]
On putting the value of \[x\] in \[(1)\], we get
\[ \Rightarrow y = {180^ \circ } - {25^ \circ }\]
\[ \Rightarrow y = {155^ \circ }\]
Therefore, the larger angle is \[{155^ \circ }\].

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 \[{90^ \circ }\].