Question:
In the given figure, AB || CD. Find the value of x.
Solution:
$A B \| C D$ and $A C$ is the transversal.
Then,
$\angle B A C+\angle A C D=180^{\circ} \quad[$ Consecutive Interior Angles $]$
$\Rightarrow 75+\angle A C D=180$
$\Rightarrow \angle A C D=105^{\circ}$
And,
$\angle A C D=\angle E C F \quad[$ Vertically-Opposite Angles $]$
$\Rightarrow \angle E C F=105^{\circ}$
We know that the sum of the angles of a triangle is 180°">180°.
$\angle E C F+\angle C F E+\angle C E F=180^{\circ}$
$\Rightarrow 105^{\circ}+30^{\circ}+x=180^{\circ}$
$\Rightarrow 135^{\circ}+x=180^{\circ}$
$\Rightarrow x=45^{\circ}$
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