Answer
Verified
395.1k+ views
Hint: Here we use the concept that both exterior angle and adjacent interior angle are supplementary to each other. So assuming the angles as two different variables we form an equation. Using the relation given in the statement form another equation. Solve both the equations to find values of the angles. Then use formula for exterior angle to find number of sides of polygon.
* If there are n sides of a polygon, then the exterior angle is given by \[\dfrac{{{{360}^ \circ }}}{n}\].
Complete step-by-step answer:
In any polygon, the exterior angle is the angle which lies outside the polygon but is on a line which is extended from the polygon. Also, the adjacent interior angle means the interior angle which is adjacent (side by side) to the exterior angle. So, we can say both adjacent interior angle and exterior angle lie on the same straight line which means they are supplementary.
Let us assume the exterior angle as \[x\] and the adjacent interior angle as \[y\].
Since, \[x\] and \[y\] are supplementary.
\[ \Rightarrow x + y = {180^ \circ }\] … (1)
Now we are given in the statement that each exterior angle is equal to one-third of its adjacent interior angle. So every \[x\] is one third of\[y\].
\[ \Rightarrow x = \dfrac{1}{3}y\]
Multiplying both sides by 3 we can write
\[ \Rightarrow 3x = y\] … (2)
Substitute the value from equation (2) in equation (1).
\[
\Rightarrow x + 3x = {180^ \circ } \\
\Rightarrow 4x = {180^ \circ } \\
\]
Divide both sides of the equation by 4
\[
\Rightarrow \dfrac{{4x}}{4} = \dfrac{{{{180}^ \circ }}}{4} \\
\Rightarrow x = {45^ \circ } \\
\]
Substituting the value in equation (2) we get
\[
\Rightarrow 3({45^ \circ }) = y \\
\Rightarrow y = {135^ \circ } \\
\]
Now we use the formula of exterior angle which is given by number of sides of a polygon i.e. exterior angle \[ = \dfrac{{{{360}^ \circ }}}{n}\]
We know the value of the exterior angle is \[{45^ \circ }\].
\[ \Rightarrow {45^ \circ } = \dfrac{{{{360}^ \circ }}}{n}\]
Cross multiply the value of \[n\] and \[{45^ \circ }\]
\[ \Rightarrow n = \dfrac{{{{360}^ \circ }}}{{{{45}^ \circ }}} = 8\]
So, the number of sides of the polygon are 8.
Note: Students can mostly get confused with the term one-third and might make wrong equation of the statement, keep in mind one third of other angle means we multiply \[\dfrac{1}{3}\]to the angle to which we are relating and equate it with the angle about which we are stating the statement.
* If there are n sides of a polygon, then the exterior angle is given by \[\dfrac{{{{360}^ \circ }}}{n}\].
Complete step-by-step answer:
In any polygon, the exterior angle is the angle which lies outside the polygon but is on a line which is extended from the polygon. Also, the adjacent interior angle means the interior angle which is adjacent (side by side) to the exterior angle. So, we can say both adjacent interior angle and exterior angle lie on the same straight line which means they are supplementary.
Let us assume the exterior angle as \[x\] and the adjacent interior angle as \[y\].
Since, \[x\] and \[y\] are supplementary.
\[ \Rightarrow x + y = {180^ \circ }\] … (1)
Now we are given in the statement that each exterior angle is equal to one-third of its adjacent interior angle. So every \[x\] is one third of\[y\].
\[ \Rightarrow x = \dfrac{1}{3}y\]
Multiplying both sides by 3 we can write
\[ \Rightarrow 3x = y\] … (2)
Substitute the value from equation (2) in equation (1).
\[
\Rightarrow x + 3x = {180^ \circ } \\
\Rightarrow 4x = {180^ \circ } \\
\]
Divide both sides of the equation by 4
\[
\Rightarrow \dfrac{{4x}}{4} = \dfrac{{{{180}^ \circ }}}{4} \\
\Rightarrow x = {45^ \circ } \\
\]
Substituting the value in equation (2) we get
\[
\Rightarrow 3({45^ \circ }) = y \\
\Rightarrow y = {135^ \circ } \\
\]
Now we use the formula of exterior angle which is given by number of sides of a polygon i.e. exterior angle \[ = \dfrac{{{{360}^ \circ }}}{n}\]
We know the value of the exterior angle is \[{45^ \circ }\].
\[ \Rightarrow {45^ \circ } = \dfrac{{{{360}^ \circ }}}{n}\]
Cross multiply the value of \[n\] and \[{45^ \circ }\]
\[ \Rightarrow n = \dfrac{{{{360}^ \circ }}}{{{{45}^ \circ }}} = 8\]
So, the number of sides of the polygon are 8.
Note: Students can mostly get confused with the term one-third and might make wrong equation of the statement, keep in mind one third of other angle means we multiply \[\dfrac{1}{3}\]to the angle to which we are relating and equate it with the angle about which we are stating the statement.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Assertion CNG is a better fuel than petrol Reason It class 11 chemistry CBSE
How does pressure exerted by solid and a fluid differ class 8 physics CBSE
Number of valence electrons in Chlorine ion are a 16 class 11 chemistry CBSE
What are agricultural practices? Define
What does CNG stand for and why is it considered to class 10 chemistry CBSE
The rate of evaporation depends on a Surface area b class 9 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
State whether the following statement is true or false class 11 physics CBSE
A night bird owl can see very well in the night but class 12 physics CBSE