Question

In figure, if $AB\parallel CD$. The value of $x$ is
(a) $122$
(b) $238$
(c) $58$
(d) $119$

Answer 1

Construction:

Draw a line $PQ$ parallel to $AB$ which is also parallel to $CD$

Since, $PQ\parallel CD$
$\Rightarrow \angle EFC=\angle FEQ={37}^{\circ }$ [Alternate interior angles]
Now, $\angle AEQ+\angle FEQ=\angle AEF$
$\Rightarrow \angle AEQ+{37}^{\circ }={95}^{\circ }$
$\Rightarrow \angle AEQ={58}^{\circ }$
Since, $PQ\parallel AB$
$\Rightarrow \angle EAB+\angle AEQ={180}^{\circ }$ [Angles on the same side of a transversal line are supplementary]
$\Rightarrow \angle EAB+{58}^{\circ }={180}^{\circ }$
$\Rightarrow \angle EAB={122}^{\circ }$
$\angle EAB+\text{Reflex}\angle EAB={360}^{\circ }$ [Complete angle]
$\Rightarrow {122}^{\circ }+(2x{)}^{\circ }={360}^{\circ }$
$\Rightarrow (2x{)}^{\circ }={360}^{\circ }-{122}^{\circ }$

$\Rightarrow (2x{)}^{\circ }={238}^{\circ }$

$\Rightarrow x^{\circ }=\frac{{238}^{\circ }}{2}$

$\Rightarrow x^{\circ }={119}^{\circ }$

$\Rightarrow x=119$

Hence, the correct answer is option (d).