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# The angles of a convex pentagon are in the ratio $2:3:5:9:11$. Find the measure of each angle.

Last updated date: 20th Jun 2024
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Hint: We are given that the angles of a convex pentagon are in the ratio $2:3:5:9:11$. Consider the angles as $2x$, $3x$, $5x$, $9x$ and $11x$. Now the polygon has five angles and therefore five sides since it is pentagon. Now we know that the sum of the angles of a pentagon is $(2n-4)\times 90{}^\circ$. We get the value of $x$. After that, you substitute the value of $x$ in $2x$, $3x$, $5x$, $9x$ and $11x$ you will get the angles.

We are given, the angles of a convex pentagon are in the ratio $2:3:5:9:11$.
Now let the angle be $2x$, $3x$, $5x$, $9x$ and $11x$.
Since, the polygon is pentagon it has five sides.
The polygon has five angles and therefore five sides.
We know that the sum of the angles of a pentagon is $(2n-4)\times 90{}^\circ$.
Now the sum of angles is $(2n-4)\times 90{}^\circ$.
Here, $n=5$,
$\Rightarrow$ $(2n-4)\times 90{}^\circ =(2\times 5-4)\times 90{}^\circ$
$\Rightarrow$ $(2n-4)\times 90{}^\circ =(10-4)\times 90{}^\circ$
$\Rightarrow$ $(2n-4)\times 90{}^\circ =(6)\times 90{}^\circ$
$\Rightarrow$ $(2n-4)\times 90{}^\circ =540{}^\circ$
Therefore, $2x+3x+5x+9x+11x=540{}^\circ$
Now simplifying we get,
$\Rightarrow$ $30x=540{}^\circ$
Now dividing whole equation by $30$ and simplifying we get,
$\Rightarrow$ $x=18{}^\circ$
So, now let us find the angles,
For $2x=2\times 18{}^\circ =36{}^\circ$
$\Rightarrow$ $3x=3\times 18{}^\circ =54{}^\circ$
$\Rightarrow$ $5x=5\times 18{}^\circ =90{}^\circ$
$\Rightarrow$ $9x=9\times 18{}^\circ =162{}^\circ$
$\Rightarrow$ $11x=11\times 18{}^\circ =198{}^\circ$
The angles of a convex pentagon are in the ratio $2:3:5:9:11$. So, the measures of each angle are $36{}^\circ ,54{}^\circ ,90{}^\circ ,162{}^\circ$ and $198{}^\circ$.

Note: A regular convex polygon is a polygon where each side is of the same length, and all the interior angles are equal and less than $180{}^\circ$. The vertices and sides are evenly spread around a central point. Also, the sum of angles of the pentagon is $(2n-4)\times 90{}^\circ$.