Question

Four angles of a polygon are $ {100^ \circ } $ each. The remaining angles are $ {160^ \circ } $ each. Find the number of sides of the polygon.

Answer 1

Hint: In the given question, we are given a polygon whose number of sides are unknown to us. We are given that four of the angles are $ {100^ \circ } $ each and the remaining of the angles of the polygon are $ {160^ \circ } $ each. The question revolves around the concepts of the polygons and their angle sum. We must know the property of the angle sum of any polygon in order to solve the problem.

Complete step-by-step answer:
In the problem given to us, we are not provided with the number of sides of the polygon. Instead, we are given that four angles of a polygon are $ {100^ \circ } $ each and the rest of the angles are $ {160^ \circ } $ each.
So, let us assume that there are n sides of the polygon.
Now, we also know that there are as many angles in a polygon as there are sides. So, there are a total of n angles in a n sided polygon.
Hence, we have four angles of n sided polygon measuring $ {100^ \circ } $ each and the remaining $ \left( {n - 4} \right) $ angles measuring $ {160^ \circ } $ each.
So, the angle sum of the polygon can be calculated as $ 4 \times {100^ \circ } + \left( {n - 4} \right) \times {160^ \circ } $ .
We also know that the angle sum of a n sided polygon is $ \left( {n - 2} \right) \times {180^ \circ } $ .
So, equating both the expression, we get the equation as,
 $ \Rightarrow 4 \times {100^ \circ } + \left( {n - 4} \right) \times {160^ \circ } = \left( {n - 2} \right) \times {180^ \circ } $
\[ \Rightarrow {400^ \circ } + {160^ \circ }\left( {n - 4} \right) = {180^ \circ }\left( {n - 2} \right)\]
Opening the brackets, we get,
\[ \Rightarrow 400 + 160n - 640 = 180n - 360\]
Now, simplifying the equation by using transposition method, we get,
\[ \Rightarrow 400 + 360 - 640 = 180n - 160n\]
\[ \Rightarrow 20n = 120\]
Dividing both sides of the equation by $ 20 $ , we get,
\[ \Rightarrow n = 6\]
Hence, the number of sides of the polygon is $ 6 $ .
So, the correct answer is “6”.

Note: So, we can find the angle sum of any polygon using the formula $ \left( {n - 2} \right) \times {180^ \circ } $ if we know the number of sides of that particular polygon. Method of transposition involves doing the exact same thing on both sides of an equation with the aim of bringing like terms together and isolating the variable or the unknown term in order to simplify the equation and finding value of the required parameter.