1. **Problem statement:** Calculate the size of angle $NPQ$ in the given polygon with points $N$, $P$, $Q$, and $R$. Given angles are $\angle R = 36^\circ$ and $\angle Q = 107^\circ$. Also, $RN = NQ$ (isosceles triangle property). 2. **Identify the triangle involved:** Since $RN = NQ$, triangle $RNQ$ is isosceles with $\angle R = 36^\circ$ and $\angle Q = 107^\circ$. 3. **Calculate the third angle in triangle $RNQ$:** $$\angle N = 180^\circ - \angle R - \angle Q = 180^\circ - 36^\circ - 107^\circ = 37^\circ$$ 4. **Use the fact that $\angle NPQ$ is adjacent to $\angle N$ and $\angle Q$:** Since $P$ lies on the polygon and $PQ$ extends outward from $Q$, $\angle NPQ$ is the external angle at $P$ adjacent to $\angle N$. 5. **Calculate $\angle NPQ$ using the external angle theorem:** The external angle equals the sum of the two opposite internal angles in the triangle. $$\angle NPQ = \angle R + \angle Q = 36^\circ + 107^\circ = 143^\circ$$ **Final answer:** $$\boxed{143^\circ}$$