Question : If the angles $P, Q$ and $R$ of $\triangle PQR$ satisfy the relation $2 \angle R-\angle P=\angle Q-\angle R$, then find the measure of $\angle R$.
Option 1: $45^{\circ}$
Option 2: $60^{\circ}$
Option 3: $50^{\circ}$
Option 4: $55^0$
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Correct Answer: $45^{\circ}$
Solution :
We know that the sum of the angles of a triangle = $180^{\circ}$
Let $\angle R = x$
Given: $2 \angle R-\angle P=\angle Q-\angle R$
⇒ $2x - \angle P = \angle Q -x$
⇒ $3x = \angle Q+\angle P$
Now, $\angle R + \angle Q + \angle P= 180^{\circ}$
⇒ $x+3x = 180^{\circ}$
⇒ $4x = 180^{\circ}$
⇒ $x= 45^{\circ}$
So, the value of $\angle R = 45^{\circ}$.
Hence, the correct answer is $45^{\circ}$.
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