Question

If $\angle DEF$ and $\angle FEG$ are supplementary. $m\angle DEF = {(9x + 1)^ \circ }$ and $m\angle FEG = {(8x + 9)^ \circ }$. How do you find the measures of both angles?

Answer 1

Hint: In this question, we need to find the values of the angles when the two angles are supplementary. Firstly, we use the definition of supplementary angles which says that ‘’Two angles are supplementary when their measure adds to ${180^ \circ }$’’. Using this we find the value of the unknown variable x. And then after finding x we substitute it in the given equation of angles and solve to obtain the values of angles in degrees.

Complete step by step answer:
Given $\angle DEF$ and $\angle FEG$ are supplementary.
Also it is mentioned that $m\angle DEF = {(9x + 1)^ \circ }$ and $m\angle FEG = {(8x + 9)^ \circ }$.
We are asked to find out the values of the angles $\angle DEF$ and $\angle FEG$.
For this we must know the value of the unknown variable x.
We use the idea that given angles are supplementary.
According to the definition, two angles are supplementary when their measure adds to ${180^ \circ }$.
So we make use of this definition to find out the value of x.
Since $\angle DEF$ and $\angle FEG$ are supplementary angles, we have,
$m\angle DEF + m\angle FEG = {180^ \circ }$
$ \Rightarrow {(9x + 1)^ \circ } + {(8x + 9)^ \circ } = {180^ \circ }$
Now we solve the equation,
$(9x + 1) + (8x + 9) = 180$
Removing the parentheses and rearranging the above equation we get,
$ \Rightarrow 9x + 8x + 1 + 9 = 180$
Combining like terms $9x + 8x = 17x$
Combining like terms $1 + 9 = 10$
Hence we get,
$ \Rightarrow 17x + 10 = 180$
Subtracting 10 from both the sides we get,
$ \Rightarrow 17x + 10 - 10 = 180 - 10$
$ \Rightarrow 17x + 0 = 170$
$ \Rightarrow 17x = 170$
Dividing throughout by 17 we get,
$ \Rightarrow \dfrac{{17x}}{{17}} = \dfrac{{170}}{{17}}$
$ \Rightarrow x = 10$
Hence the value of the unknown variable is $x = 10$.
Now substituting this value in the equation of angles we get the value of the two supplementary angles.
We have $\angle DEF = {(9x + 1)^ \circ }$
$ \Rightarrow \angle DEF = {(9 \cdot 10 + 1)^ \circ }$
$ \Rightarrow \angle DEF = {(90 + 1)^ \circ }$
$ \Rightarrow \angle DEF = {91^ \circ }$
Now we find the other angle.
We have $\angle FEG = {(8x + 9)^ \circ }$
$ \Rightarrow \angle FEG = {(8 \cdot 10 + 9)^ \circ }$
$ \Rightarrow \angle FEG = {(80 + 9)^ \circ }$
$ \Rightarrow \angle FEG = {89^ \circ }$
Hence the value of the supplementary angles are given by
$\angle DEF = {91^ \circ }$ and $\angle FEG = {89^ \circ }$.

Note: Students must know the basic definitions related to angles.
Two angles are called complementary when their measure adds to ${90^ \circ }$.
Two angles are called supplementary when their measure adds to ${180^ \circ }$.
If we are asked to find the measure of an angle when it is given by an equation with an unknown variable, we need to solve the equation and find the value of the variable. Then we substitute it back in the given equation, to obtain the required value of the angle.