ABCD is a parallelogram. BC is produced to Q such that BC = CQ. Then

Options

A. (a) area (`Delta BCP)` = area (`Delta DPQ)`
B. (b) area `(Delta BCP) gt` area (`Delta DPQ)`
C. (c) area `(Delta BCP) lt` area `(Delta DPQ)`
D. (d) area `(Delta BCP)` + area (`Delta DPQ)` = area `(Delta BCD)`

Answer

To solve the problem, we need to analyze the given information about the parallelogram ABCD and the extension of line BC to point Q such that BC = CQ. We will compare the areas of triangles ABC and DCQ.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- ABCD is a parallelogram.
- BC is extended to point Q such that BC = CQ.

2. **Understand the Properties of Parallelograms:**
- In a parallelogram, opposite sides are equal, and opposite angles are equal.
- Additionally, consecutive angles are supplementary.

3. **Analyze Triangles ABC and DCQ:**
- We need to compare the areas of triangle ABC and triangle DCQ.
- Since BC = CQ, we can denote the length of BC as x. Therefore, CQ is also x.

4. **Identify Corresponding Angles:**
- Since ABCD is a parallelogram, we know:
- Angle ABC = Angle DCQ (corresponding angles).
- Angle A = Angle D (alternate interior angles).

5. **Apply the Angle-Side-Angle (ASA) Congruence Criterion:**
- We have:
- Angle ABC = Angle DCQ (from step 4).
- Angle A = Angle D (from step 4).
- Side BC = Side CQ (given).
- Therefore, by the ASA criterion, triangle ABC is congruent to triangle DCQ.

6. **Conclude About the Areas:**
- Since triangle ABC is congruent to triangle DCQ, their areas are equal.
- Thus, Area of triangle ABC = Area of triangle DCQ.

### Final Answer:
The area of triangle ABC is equal to the area of triangle DCQ.