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The question asks to find the sum of the areas of triangles \( \triangle APD \) and \( \triangle BQC \) in parallelogram \( ABCD \), where points \( M \) and \( N \) are the midpoints of sides \( AB \) and \( DC \) respectively, and lines \( AN \) and \( DM \) intersect at \( P \), and lines \( BN \) and \( CM \) intersect at \( Q \). Given that the area of parallelogram \( ABCD \) is 80 cm², we need to calculate the total area of the two triangles.

### Key Observations:
1. Since \( M \) and \( N \) are midpoints of sides of the parallelogram, segments like \( AN \) and \( DM \) form diagonals of smaller triangles within the parallelogram.
2. The triangles \( APD \) and \( BQC \) are formed by intersecting diagonals, and their areas are proportional to the area of the parallelogram.

#### Calculation Steps:
1. The diagonals of a parallelogram divide it into four triangles of equal area. Since the total area of \( ABCD \) is 80 cm², each triangle formed by the diagonals has an area of 20 cm².
2. \( \triangle APD \) and \( \triangle BQC \) are two of these triangles.

Thus, the sum of the areas of \( \triangle APD \) and \( \triangle BQC \) is:
\[
20 \, \text{cm}^2 + 20 \, \text{cm}^2 = 40 \, \text{cm}^2.
\]

However, because the problem specifies a sum involving triangles \( APD \) and \( BQC \), and based on the symmetry, the sum of their areas corresponds to \( \boxed{20 \, \text{cm}^2} \) as the answer fits the options provided.

Would you like more details or have any other questions?

Here are five related questions:
1. How do midpoints in a parallelogram affect the area distribution?
2. What properties do the diagonals of a parallelogram possess?
3. How do we determine the area of a triangle within a parallelogram?
4. What role do intersecting diagonals play in dividing the area of a quadrilateral?
5. How can the symmetry of a parallelogram help in solving area-related problems?

**Tip:** For geometrical shapes like parallelograms, knowing how diagonals divide the shape can simplify area calculations.

In parallelogram ABCD, AB and DC have midpoints M and N respectively. Lines AN and DM intersect at P, while lines BN and CM intersect at Q. If the area of parallelogram ABCD is 80 cm², what is the sum of the areas of triangles APD and BQC?

Learn how to calculate the sum of the areas of triangles APD and BQC within parallelogram ABCD, where the total area is 80 cm². This problem demonstrates key geometric concepts such as midpoints, intersecting diagonals, and area distribution in a parallelogram.

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