Question

find the largest angle of $\triangle pqr$. assume that $y$ is a positive number.

Explanation:

Step1: Compare side lengths

Since $y>0$, $15y > 5y$, so $PQ > QR$. Also, by the triangle inequality, the third side $PR$ satisfies $15y - 5y < PR < 15y + 5y$, so $10y < PR < 20y$.

Step2: Relate sides to angles

In a triangle, the largest angle is opposite the longest side. We know $\angle R$ is opposite $PQ=15y$, $\angle P$ is opposite $QR=5y$, and $\angle Q=96^\circ$ is opposite $PR$. Since $PR < 20y$, but we already have $\angle Q=96^\circ$, which is greater than $90^\circ$. For any triangle, there can be at most one obtuse angle (greater than $90^\circ$), so $\angle Q$ is the largest angle.

Answer:

$\angle Q$ (or $96^\circ$)