Let's analyze and solve the problem step by step.

Given:

• $\triangle ABC$ where $BD$ is the altitude, and $H$ is the orthocenter.
• The perpendicular bisector of $HD$ meets line $BCD$ at points $P$ and $Q$.
• We need to find $\angle APB + \angle AQB$.

Solution:

1. Understanding the Geometry:

   • Since $H$ is the orthocenter, $BD$ is perpendicular to $AC$.
   • $P$ and $Q$ lie on the perpendicular bisector of $HD$, which makes $P$ and $Q$ equidistant from $H$ and $D$.

2. Key Concept:

   • Since $P$ and $Q$ are on the perpendicular bisector, $\triangle HPD$ and $\triangle HQD$ are isosceles triangles, with $HP = PD$ and $HQ = QD$.
   • The configuration suggests that $P$ and $Q$ might be symmetric with respect to line $AC$, indicating some symmetry in the angles we need to find.

3. Utilizing Symmetry:

   • By the properties of perpendicular bisectors and symmetry, $\triangle APB$ and $\triangle AQB$ should be congruent.
   • This implies that $\angle APB = \angle AQB$.
   • Let $\angle APB = \alpha$. Therefore, $\angle AQB = \alpha$.

4. Sum of Angles:

   • $\angle APB + \angle AQB = \alpha + \alpha = 2\alpha$.
   • Since $APBQ$ is a cyclic quadrilateral (as $P$ and $Q$ lie on the perpendicular bisector and symmetry), $\angle APB$ and $\angle AQB$ add up to $180^\circ$.

Thus, the sum $\angle APB + \angle AQB = 180^\circ$.

Conclusion:

$\boxed{\angle APB + \angle AQB = 180^\circ}$

Would you like further details or clarifications on this solution? Here are some related questions:

1. Why does the perpendicular bisector ensure symmetry between points $P$ and $Q$?
2. What properties of the orthocenter are crucial for this problem?
3. How can we prove that $APBQ$ is a cyclic quadrilateral?
4. What role does symmetry play in determining the angles in this problem?
5. Can this method be generalized to other configurations involving perpendicular bisectors and orthocenters?

Tip: Remember that in many geometry problems involving symmetry, congruent triangles can reveal equal angles or distances, which can simplify the solution process.