1. **State the problem:** We need to find the measure of angle $\angle N$ in an isosceles triangle $\triangle NPM$ where sides $NM$ and $MP$ are equal. 2. **Identify given information:** Since $NM = MP$, the triangle is isosceles with angles opposite these sides equal. Therefore, $\angle N = \angle P$. 3. **Express angles in terms of $y$:** - $\angle N = 6y^\circ$ - $\angle P = (8y - 16)^\circ$ 4. **Set the angles equal:** $$6y = 8y - 16$$ 5. **Solve for $y$:** $$6y - 8y = -16$$ $$-2y = -16$$ $$y = 8$$ 6. **Find $\angle N$ by substituting $y=8$:** $$\angle N = 6y = 6 \times 8 = 48^\circ$$ 7. **Verify the triangle angle sum:** The third angle $\angle M$ can be found by: $$\angle M = 180^\circ - (\angle N + \angle P) = 180 - (48 + 48) = 84^\circ$$ **Final answer:** $$\boxed{48^\circ}$$