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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.

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Last updated date: 20th Jun 2024
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Answer
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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.

Complete step-by-step answer:
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 $.

Additional information:
A polygon is called a convex polygon if all the interior angles are less than 180°. Regularly, a polygon is firmly convex, if each line segment with two nonadjacent vertices of the polygon is strictly internal to the polygon but on its endpoints. Each non-fragment triangle is definitely convex. The measures of the interior angles in a convex polygon are strictly less than 180 degrees. Convex polygons are the exact inverse of concave polygons. The vertices of a convex polygon always point outwards.

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 $.