To prove that ar(APD) = ar(BQC), we need to show that the triangles APD and BQC have equal areas.

Let's break down the proof into different steps:

Step 1: Establishing the Parallelogram ABCD
- Given that P and Q are two points lying on the sides DC and AD of parallelogram ABCD, we can assume that ABCD is a parallelogram.
- In a parallelogram, opposite sides are parallel and equal in length.

Step 2: Drawing Diagonals
- Draw diagonal AC, which divides the parallelogram into two triangles, namely ADC and ABC.
- Draw diagonal BD, which divides the parallelogram into two triangles, namely BCD and ABD.

Step 3: Proving Triangles ADC and ABC are Congruent
- In a parallelogram, opposite sides are parallel, and hence, the height or perpendicular distance between them is the same.
- Therefore, the height of triangle ADC is equal to the height of triangle ABC.
- The base AD is common in both triangles.
- By the formula for the area of a triangle (Area = 1/2 * base * height), we can conclude that the areas of triangles ADC and ABC are equal.

Step 4: Proving Triangles BCD and ABD are Congruent
- Similar to step 3, we can prove that the areas of triangles BCD and ABD are equal.

Step 5: Proving Triangles APD and BQC are Congruent
- Since triangles ADC and ABC are congruent, the corresponding parts of these triangles are also congruent.
- Therefore, angles DAP and CBQ are congruent, and angles APD and BQC are congruent.
- The base AD is equal to the base BC.
- By the formula for the area of a triangle (Area = 1/2 * base * height), we can conclude that the areas of triangles APD and BQC are equal.

Step 6: Conclusion
- We have proved that the areas of triangles APD and BQC are equal, which implies that ar(APD) = ar(BQC).

In summary, by establishing the properties of a parallelogram and using congruence of triangles, we can prove that the areas of triangles APD and BQC are equal.