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(a) 4

(b) 6

(c) 8

(d) 10

Answer
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\[\text{Sum of all interior angles}=\left( n-2 \right){{180}^{\circ }}\]

So, we will get,

\[\text{n}\times \text{13}{{\text{5}}^{\circ }}=\left( n-2 \right){{180}^{\circ }}\]

We will solve for n and then get the number of sides of the polygon.

We are given that each interior angle is \[{{135}^{\circ }},\] we have to find the number of sides of the polygon. Let’s start with the assumption that the number of sides be n, this means that we have n sided regular polygon whose interior angle is \[{{135}^{\circ }}.\]

We know that their relation between the interior angle and the number of sides of the polygon is given as,

\[\text{Sum of all interior angles}=\left( n-2 \right)\times {{180}^{\circ }}\]

Our n sided regular polygon has n interior angles with each of 135 degrees. So, the sum of all the interior angles will be

\[\text{Sum of all interior angles}=n\times {{135}^{\circ }}\]

So, as we have the sum of all the interior angles given as \[\left( n-2 \right)\times {{180}^{\circ }},\] so, we get,

\[\Rightarrow n\times {{135}^{\circ }}=\left( n-2 \right)\times {{180}^{\circ }}\]

Now, opening the brackets, we get,

\[\Rightarrow 135n=180n-{{360}^{\circ }}\]

Now simplifying the term, we get,

\[\Rightarrow 136n-180n=-360\]

Solving for n, we get,

\[\Rightarrow -45n=-360\]

Dividing both sides by – 45, we get,

\[n=\dfrac{-360}{-45}=8\]

Therefore, we get the number of sides that the polygon has as 8.

\[\text{n}\times \text{Interior Angles}=\left( n-2 \right)\times {{180}^{\circ }}\]

As the interior angle is \[{{135}^{\circ }},\] so we get,

\[n\times {{135}^{\circ }}=\left( n-2 \right){{180}^{\circ }}\]

(a) If we take n = 4,

\[n\times {{135}^{\circ }}=4\times {{135}^{\circ }}={{540}^{\circ }}\]

While,

\[\left( n-2 \right){{180}^{\circ }}=\left( 4-2 \right){{180}^{\circ }}={{360}^{\circ }}\]

Both are not equal and hence (a) is not the right option.

(b) If we take n = 6,

\[n\times {{135}^{\circ }}=6\times {{135}^{\circ }}={{810}^{\circ }}\]

While,

\[\left( n-2 \right){{180}^{\circ }}=\left( 6-2 \right){{180}^{\circ }}={{720}^{\circ }}\]

Both are not equal and hence (b) is not the right option.

(c) If we take n = 10,

\[n\times {{135}^{\circ }}=10\times {{135}^{\circ }}={{1350}^{\circ }}\]

While,

\[\left( n-2 \right){{180}^{\circ }}=\left( 10-2 \right){{180}^{\circ }}={{1440}^{\circ }}\]

Both are not equal and hence (c) is not the right option.