Question : In $\Delta PQR,$ $\angle P : \angle Q : \angle R = 1: 3 : 5$, what is the value of $\angle R - \angle P$?
Option 1: $30^\circ$
Option 2: $80^\circ$
Option 3: $45^\circ$
Option 4: $60^\circ$
Correct Answer: $80^\circ$
Solution :
The sum of the angles in a triangle is $180^\circ$.
Given the ratio of the angles $\angle P: \angle Q: \angle R = 1: 3: 5$.
Let the common ratio be $x$.
Such that, $\angle P = x$, $\angle Q = 3x$, and $\angle R = 5x$.
$⇒x + 3x + 5x = 180^\circ$
$⇒9x = 180^\circ$
$⇒x = 20^\circ$
$\angle P = 20^\circ$, $\angle Q = 60^\circ$, and $\angle R = 100^\circ$
The value of $\angle R - \angle P=100^\circ - 20^\circ = 80^\circ$.
Hence, the correct answer is $80^\circ$.
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