Question
In Fig. $PQ || RS, TR || QU$ and $\angle\text{PTR} = 42^\circ$. Find $\angle\text{QUR}$.
✓
Answer
Since, $PQ$ and $RS$ are parallel and $TR$ is transversal.
Therefore, $\angle\text{PTR}=\angle\text{TRU} [$Alternate interior angles$]$
$\Rightarrow\angle\text{TRU}=42^\circ$
Now, $TR$ is parallel to $QU$ and $RS$ is transversal.
Therefore, $\angle\text{TRU}+\angle\text{RUQ}=180^\circ [$Consecutive interior angles$]$
$\Rightarrow\angle42^\circ+\angle\text{RUQ}=180^\circ$
$\Rightarrow\angle\text{ROQ}=180^\circ-42^\circ=138^\circ$
Therefore, $\angle\text{PTR}=\angle\text{TRU} [$Alternate interior angles$]$
$\Rightarrow\angle\text{TRU}=42^\circ$
Now, $TR$ is parallel to $QU$ and $RS$ is transversal.
Therefore, $\angle\text{TRU}+\angle\text{RUQ}=180^\circ [$Consecutive interior angles$]$
$\Rightarrow\angle42^\circ+\angle\text{RUQ}=180^\circ$
$\Rightarrow\angle\text{ROQ}=180^\circ-42^\circ=138^\circ$
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