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# In figure, if PQ || ST, $$\angle{PQR}$$ = $$110^\circ$$ and $$\angle{RST}$$ = $$130^\circ$$ , find $$\angle{QRS}$$ .

Construction : Draw a line parallel to ST through R.

As it is given that,
PQ || ST,$$\angle{PQR}$$ = $$110^\circ$$ and $$\angle{RST}$$ = $$130^\circ$$
We can also say that, AB || PQ || ST

$$\because$$ $$\angle{PQR}$$ + $$\angle{QRA}$$ = $$180^\circ$$
....(interior angles on the same side of transversal)
$$\therefore$$ $$110^\circ$$ + $$\angle{QRA}$$ = $$180^\circ$$
$$\Rightarrow$$ $$\angle{QRA}$$ = $$70^\circ$$

$$\because$$ $$\angle{ARS}$$ = $$130^\circ$$ ....(Alternate Interior angle)
As, $$\angle{RST}$$ = $$130^\circ$$
Now, so as to find $$\angle{QRS}$$, We have,
$$\angle{ARS}$$ = $$\angle{ARQ}$$ + $$\angle{QRS}$$
$$\Rightarrow$$ $$130^\circ$$ = $$70^\circ$$ + $$\angle{QRS}$$
$$\Rightarrow$$ $$\angle{QRS}$$ = $$60^\circ$$