Question
In the given figure, $AB\ ||\ CD$, $\angle\text{BAE}=65^\circ$ and $\angle\text{OEC}=20^\circ.$ Find $\angle\text{ECO}.$ 
✓
Answer
$AB\ ||\ CD$ and $AE$ is the transversal.
$\Rightarrow\angle\text{BAE}=\angle\text{DOE}$ (corresponding angles)
$\angle\text{DOE}=65^\circ$ Now,
$\angle\text{DOE}+\angle\text{COE}=180^\circ$ (linear pair)
$\Rightarrow65^\circ+\angle\text{COE}=180^\circ$
$\Rightarrow\angle\text{COE}=115^\circ$ In
$\triangle\text{OCE},$ by angle sum property,
$\angle\text{OEC}+\angle\text{ECO}+\angle\text{COE}=180^\circ$
$\Rightarrow20^\circ+\angle\text{ECO}+115^\circ=180^\circ$
$\Rightarrow\angle\text{ECO}=45^\circ$
$\Rightarrow\angle\text{BAE}=\angle\text{DOE}$ (corresponding angles)
$\angle\text{DOE}=65^\circ$ Now,
$\angle\text{DOE}+\angle\text{COE}=180^\circ$ (linear pair)
$\Rightarrow65^\circ+\angle\text{COE}=180^\circ$
$\Rightarrow\angle\text{COE}=115^\circ$ In
$\triangle\text{OCE},$ by angle sum property,
$\angle\text{OEC}+\angle\text{ECO}+\angle\text{COE}=180^\circ$
$\Rightarrow20^\circ+\angle\text{ECO}+115^\circ=180^\circ$
$\Rightarrow\angle\text{ECO}=45^\circ$
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