An isosceles triangle PQR with PQ = QR, implying base angles are equal. Let the equal angles be x and the third angle be y. The condition is x = 15 + (1/4)y.

Use the option elimination by substituting: if Q = 120, then base angles = (180-120)/2 = 30. Check if 30 = 15 + 120/4. Since 30 = 15 + 30 = 45 is false, re-evaluate: if base angles are equal to 30, Q is 120. Check 30 = 15 + 120/4 → 30 = 15 + 30 (False). Wait, let equal angles be $x$. $2x+y=180$. $x = 15 + y/4$. Substitute $y = 180 - 2x$: $x = 15 + (180-2x)/4 \implies x = 15 + 45 - 0.5x \implies 1.5x = 60 \implies x = 40$. Thus $y = 180 - 80 = 100$.

Students often assume the equal angles are adjacent to the vertex Q, but the problem states PQ = QR, meaning the equal angles are $\angle P$ and $\angle R$, and the vertex angle is $\angle Q$.

This problem is a classic application of linear systems in geometry. Always verify the sum of internal angles equals 180 degrees to catch arithmetic errors.