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The angles of a quadrilateral are in the ratio 2 : 3 : 5 : 8. Find the smallest angle of the quadrilateral.

Last updated date: 13th Jun 2024
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Hint: We will first discuss the fact that sum of angles of quadrilateral is ${360^ \circ }$. Then, we will just assume the angle and find all other angles using the ratio and sum them all up and equate to ${360^ \circ }$, we will have with us, the measures of the smallest angle.

Complete step-by-step answer:
We will use the fact that the sum of angles of a quadrilateral is ${360^ \circ }$.
Sum of angles of quadrilateral is ${360^ \circ }$ ………(1)
Now, we have the ratio of angles as 2 : 3 : 5 : 8.
Let the first angle be $2x$, so then according to the ratio, the angles will be $2x,3x,5x,8x$.
Now using (1), we will have:-
$2x + 3x + 5x + 8x = {360^ \circ }$
Simplifying the LHS, we will have:-
$18x = {360^ \circ }$
Taking the 15 from multiplication in LHS to division in RHS, we will have:-
\[x = \dfrac{{{{360}^ \circ }}}{{18}} = \dfrac{{{{40}^ \circ }}}{2} = {20^ \circ }\]
Hence, $x = {20^ \circ }$.
So, the angles will be ${40^ \circ },{60^ \circ },{100^ \circ },{160^ \circ }$.
$\therefore $ The smallest angle of the quadrilateral = ${40^\circ }$

Note: The students might leave the answer part after finding the value of x, but they must remember that they are asked about the measure of angles, not the value to solve them. So, we must complete our answer by substituting the value of x in the required angle.