In figure, if $AB\parallel CD,CD\parallel EF$ and $y:z = 3:7$, find $x$.

$A){\text{ }}x = 186_{}^\circ$
$B){\text{ }}x = 156_{}^\circ$
$C){\text{ }}x = 126_{}^\circ$
$D){\text{ }}x = 106_{}^\circ$

Hint: Let us assume that $y = 3a$ and $z = 7a$ to solve this question. After that we have to apply the concept that the sum of the interior angles on the same side of the transversal equals to $180_{}^\circ$.

Complete step-by-step answer:
It is given in the question that $AB\parallel CD,CD\parallel EF$ therefore we can consider that $AB\parallel EF$ and we have considered that $y = 3a$ and $z = 7a$
Therefore we can write $x = z$$ = 7a$ since we have assumed that $z = 7a$
Now, by applying the concept that the sum of the angles on the same side of the transversal equals to $180_{}^\circ$.
So we can write,$\angle x + \angle y = 180_{}^\circ$…..$(i)$
Putting the values of $x$ and $y$ in equation$(i)$we get-
$7a + 3a = 180_{}^\circ$
By doing addition we get-
$10a = 180_{}^\circ$
By division we get-
So, $a = \dfrac{{180_{}^\circ}}{{10}} = 18_{}^\circ$
Therefore, the value of $x = 7a$
$ = 7 \times 18_{}^\circ$
On multiplying the term we get
$ = 126_{}^\circ$

Hence, the correct option is $C$.

Note: The conditions which are formed when a pair of parallel lines is intersected by a transversal must be kept in mind as most of the students make mistakes in the calculation.
Transversal is a straight line which intersects a pair of parallel lines at distinct points. The angles that are formed when a transversal intersects a pair of two or more lines are alternate angles and corresponding angles.
Where a transversal intersects two parallel lines, the corresponding angles become equal, the vertically opposite angles are also equal, the alternate exterior and interior angles are also equal and the sum of the interior angles on the same side of the transversal which is known as co-interior angles equals to \[180_{}^\circ\].