Question 122
Points $P$ and $Q$ lie inside acute triangle $ABC$ such that $\angle PAB = \angle QAC$ and $\angle PBA = \angle QBC$. Point $D$ lies on side $BC$. Prove that
$$
\angle DPC + \angle APB = 180^{\circ} \quad \text{if and only if} \quad \angle DQB + \angle AQC = 180^{\circ}.
$$
$$
\angle DPC + \angle APB = 180^{\circ} \quad \text{if and only if} \quad \angle DQB + \angle AQC = 180^{\circ}.
$$