The angles of a quadrilateral are in the ratio $2:4:5:7$ . Find the angles.

Hint: We will first discuss the fact that the sum of angles of quadrilaterals 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 ${360^ \circ }$ , after that we will find all the angles.

Complete step-by-step solution:
We will use the fact that 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 $2:4:5:7$ .
Let the first angle be $2x$ , so then according to the ratio, the angles will be $2x,4x,5x,7x$ .
Now using (1), we will have
$2x + 4x + 5x + 7x = {360^ \circ }$
Simplifying the LHS, we will have
$ \Rightarrow 18x = {360^ \circ }$
Now we divide both sides by $18$ and we get
$ \Rightarrow x = \dfrac{{{{360}^ \circ }}}{{18}}$
Simplifying and we get
So, the angles will be $2 \times {20^ \circ },4 \times {20^ \circ },5 \times {20^ \circ }$ and $7 \times {20^ \circ }$
Therefore the angles are ${40^ \circ },{80^ \circ },{100^ \circ }$ and ${140^ \circ }$.

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.