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If the angles of a pentagon are in the ratio $7:8:11:13:15$ , find the angles.

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Last updated date: 24th Feb 2024
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IVSAT 2024
Answer
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Hint: To obtain the angles of a pentagon we will use formula for sum of measure of all interior angles of a polygon. Firstly as we know pentagon is a five sided polygon so we will put that value in the formula and get the sum of its all angles. Then we let the fraction given and put the sum of them equal to the sum of all angles. Finally we will simplify the obtained equation and get the value of all the angles and our desired answer.

Complete step by step answer:
We have to find the angles of a pentagon which are in ratio
$7:8:11:13:15$…..$\left( 1 \right)$
As we know that
Sum of measures of all interior angles of polygon $=\left( 2n-4 \right)\times {{90}^{\circ }}$……$\left( 2 \right)$
A pentagon has five sides so
$n=5$
Put $n=5$ in equation (2) we get,
$\begin{align}
  & \Rightarrow \left( 2\times 5-4 \right)\times {{90}^{\circ }} \\
 & \Rightarrow \left( 10-4 \right)\times {{90}^{\circ }} \\
 & \Rightarrow 6\times {{90}^{\circ }} \\
 & \Rightarrow {{540}^{\circ }} \\
\end{align}$
So sum of all interior angle of a pentagon is ${{540}^{\circ }}$
Next, let us assume from equation (1) that the angle of the pentagon are as below:
$7x,8x,11x,13x,15x$…..$\left( 3 \right)$
So the sum of above angle will be equal to ${{540}^{\circ }}$
$\begin{align}
  & 7x+8x+11x+13x+15x={{540}^{\circ }} \\
 & \Rightarrow 54x={{540}^{\circ }} \\
\end{align}$
$\begin{align}
  & \Rightarrow x=\dfrac{{{540}^{\circ }}}{54} \\
 & \therefore x={{10}^{\circ }} \\
\end{align}$
Now put the above value in equation (3) and simplify as below:
$\begin{align}
  & 7x=7\times {{10}^{\circ }} \\
 & 7x={{70}^{\circ }} \\
\end{align}$
$\begin{align}
  & 8x=8\times {{10}^{\circ }} \\
 & 8x={{80}^{\circ }} \\
\end{align}$
$\begin{align}
  & 11x=11\times {{10}^{\circ }} \\
 & 11x={{110}^{\circ }} \\
\end{align}$
$\begin{align}
  & 13x=13\times {{10}^{\circ }} \\
 & 13x={{130}^{\circ }} \\
\end{align}$
$\begin{align}
  & 15x=15\times {{10}^{\circ }} \\
 & 15x={{150}^{\circ }} \\
\end{align}$
Hence the angles of the pentagon are ${{70}^{\circ }},{{80}^{\circ }},{{110}^{\circ }},{{130}^{\circ }},{{150}^{\circ }}$

Note: A polygon is a figure that is made up of a finite number of straight line segments that are connected to form a closed polygon chain. The segments are known as the edges or the sides and the point at which two edges meet is known as the vertices or the corners of the polygon. The sum of interior angles of the polygon is calculated by the formula $\left( 2n-4 \right)\times {{90}^{\circ }}$