Question
In the given figure, $\angle\text{BAD}=75^\circ,\angle\text{DCF}=\text{x}^\circ$ and $\angle\text{DEF}=\text{y}^\circ.$ Find the values of $x$ and $y$. 
✓
Answer
We know that if one side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle $\text{i.e.},\angle\text{BAD}=\angle\text{DCF}=75^\circ$
$\Rightarrow\ \angle\text{DCF}=\text{x}=75^\circ$ Again, the sum of opposite angles in a cyclic quadrilateral is $180^\circ$.
Thus, $\angle\text{DCF}=\angle\text{DEF}=180^\circ$
$\Rightarrow\ 75^\circ+\text{y}=180^\circ$
$\Rightarrow\ \text{y}=(180^\circ-75^\circ)=105^\circ$
Hence, $\text{x}=75^\circ$ and $\text{y}=105^\circ.$
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