Question
In the given parallelogram $YOUR$, $\angle\text{RUO}=120^\circ\text{and}\ \text{OY}$ is extended to point $S$ such that $\angle\text{SRY}=50^\circ.\text{Find}\angle\text{YSR}.$
✓
Answer
Given, $\angle\text{RUO}=120^\circ\ \text{and}\ \angle\text{SRY}=50^\circ$
$\angle\text{RYO}=\angle\text{SYR}=180^\circ-\text{RYO}$
Now, $\angle\text{SYR}=180^\circ-\angle\text{RYO}$
$=180^\circ-120^\circ=60^\circ$
$\text{In}\triangle\text{SRY},$
By the angle sum property of a triangle, $\angle\text{SYR}+\angle\text{RYS}+\angle\text{YSR}=180^\circ$
$\Rightarrow50^\circ+60^\circ+\angle\text{YSR}=180^\circ$
$\Rightarrow\angle\text{YSR}=180^\circ-(50^\circ+60^\circ)=70^\circ.$
$\angle\text{RYO}=\angle\text{SYR}=180^\circ-\text{RYO}$
Now, $\angle\text{SYR}=180^\circ-\angle\text{RYO}$
$=180^\circ-120^\circ=60^\circ$
$\text{In}\triangle\text{SRY},$
By the angle sum property of a triangle, $\angle\text{SYR}+\angle\text{RYS}+\angle\text{YSR}=180^\circ$
$\Rightarrow50^\circ+60^\circ+\angle\text{YSR}=180^\circ$
$\Rightarrow\angle\text{YSR}=180^\circ-(50^\circ+60^\circ)=70^\circ.$
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