The problem asks to find the measure of $\angle PDQ$ in the given figure.

Analyzing the Figure:

• $\angle A$ is a right angle ($90^\circ$) because it is marked as such.
• $\angle C$ is also a right angle ($90^\circ$) because it is marked as such.
• $P$, $Q$, and $D$ are points on the sides of the triangles.

Steps to Solve:

1. Identify Key Angles:

   • $\angle PDQ$ is the angle we need to find.
   • $\angle APD$ and $\angle CPD$ are supplementary since they are angles on a straight line at point $P$.

2. Use the Triangle Angle Sum Property:

   • In $\triangle APC$, the sum of the angles should be $180^\circ$.
   • Since $\angle APC = 90^\circ$, the sum of the other two angles, $\angle PAD + \angle ACP$, should be $90^\circ$.

3. Consider the Parallel Lines:

   • If $AB$ is parallel to $CD$, then $\angle PAD = \angle DQC$ (corresponding angles).

4. Find $\angle PDQ$:

   • If $\angle ACP$ is known, you can use that to find $\angle PDQ$ using the supplementary angle concept.

To provide a more precise solution, I would need the relationship or angle values between $\angle APD$, $\angle CPD$, and other relevant angles.

Without Additional Information:

Assuming the common angle relationships:

• If the triangles are isosceles or if specific angles are marked, please provide that or consider the relevant parallel and supplementary angle relationships.

Would you like a more detailed explanation on any specific part, or do you have the values for the angles? Let me know if you have more details or if I should make any specific assumptions to solve the problem.

Relative Questions:

1. How can parallel lines help in determining unknown angles in geometric figures?
2. What are the properties of corresponding angles when two parallel lines are cut by a transversal?
3. How does the Triangle Angle Sum Property assist in finding unknown angles?
4. What is the significance of supplementary angles in solving geometry problems?
5. How can you verify if two lines are parallel based on angle measurements in a figure?

Tip:

Always look for parallel lines, right angles, and use the properties of triangles (like the sum of angles) to help determine unknown angles.