In Fig. 6.31, if $ \mathrm{PQ} \| \mathrm{ST}, \angle \mathrm{PQR}=110^{\circ} $ and $ \angle \mathrm{RST}=130^{\circ} $, find $ \angle \mathrm{QRS} $. [Hint : Draw a line parallel to ST through point R.] "
Given: $PQ \parallel ST$, $\angle PQR=130^o$.
To do:
We have to find $\angle QRS$.
Solution:
Let us draw a line parallel to $ST$ through the point $R$ and name it $UV$.
We know that,
The angles on the same side of the transversal are equal to $180^o$.
Therefore,
$\angle RST+\angle SRV=180^o$
This implies,
$\angle SRV=180^o-130^o$ (Since, $\angle S=130^o$)
We get,
$\angle SRV=50^o$
In the similar way, we get,
$\angle PQR+\angle QRU=180^o$
This implies,
$\angle QRU=180^o-110^o$ (Since,$\angle Q=110^o$)
We get,
$\angle QRU=70^o$
Therefore,
$\angle QRU+\angle QRS+\angle SRV=180^o$ (Since the sum of the measures of the angles in linear pairs is always $180^o$)
Therefore by substituting the values we get,
$\angle QRS=180^o-70^o-50^o$
$\angle QRS=180^o-120^o$
$\angle QRS=60^o$
Therefore, $\angle QRS=60^o$.
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