Question : The side QR of an equilateral triangle PQR is produced to the point S in such a way that QR = RS and P is joined to S. Then the measure of $\angle PSR$ is:
Option 1: $30^{\circ}$
Option 2: $15^{\circ}$
Option 3: $60^{\circ}$
Option 4: $45^{\circ}$
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Correct Answer: $30^{\circ}$
Solution :
Given: $PQR$ is an equilateral triangle and $QR = RS$
$⇒\angle PRS = 180^{\circ} - 60^{\circ} = 120^{\circ}$
We have, $PR = RQ$ and $RQ = RS$, it follows that $RS = PR$
$\therefore \angle PSR = \angle RPS$
The sum of angles in a triangle is $180^{\circ}$.
$⇒\angle PSR + \angle RPS +\angle PRS= 180^{\circ}$
$⇒2\angle PSR =180^{\circ}-\angle PRS=180^{\circ}-120^{\circ}$
$⇒2\angle PSR = 60^{\circ}$
$\therefore \angle PSR= 30^{\circ}$
Hence, the correct answer is $30^{\circ}$.
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