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Which congruency theorem can be used to prove that △ABD ≅ △DCA?

A.     SSS

B.     ASA

C.     SAS

D.     AAS

To analyze which congruency theorem can be used to prove that triangles △ABD and △DCA are congruent, let's break down the figure and apply geometric reasoning.

### Step-by-Step Analysis:

1. **Opposite sides of a rectangle**: The diagram shows a quadrilateral that resembles a rectangle (due to the right angles at two corners). In a rectangle:
   - Opposite sides are equal.
   - The diagonals are equal and bisect each other.

2. **Triangles to be proved congruent**: We are focusing on triangles △ABD and △DCA.
   - **AB = DC**: These sides are opposite sides of the rectangle, so they are equal.
   - **AD = AD**: This side is common to both triangles.
   - **BD = AC**: The diagonals of a rectangle are equal, so these two sides are also equal.

### Conclusion:
We have three pairs of corresponding sides that are equal between triangles △ABD and △DCA. Therefore, the **SSS (Side-Side-Side)** congruence theorem can be applied here.

The correct answer is:

**A. SSS**

Would you like a deeper explanation of any step? Here are some related questions to explore:

1. What properties do diagonals of rectangles have?
2. How does the SSS congruence theorem work in general?
3. Could the SAS or ASA theorems be used in this situation?
4. Why is the right angle irrelevant in proving congruence here?
5. How would the proof change if this were not a rectangle but another type of quadrilateral?

**Tip**: When dealing with shapes like rectangles, always remember that diagonals have special properties—they bisect each other and are of equal length.

Which congruency theorem can be used to prove that △ABD ≅ △DCA?

A. SSS
B. ASA
C. SAS
D. AAS

Learn how to prove that triangles ABD and DCA are congruent using the Side-Side-Side (SSS) congruence theorem. This problem demonstrates the geometric properties of rectangles and their diagonals, and it's perfect for students in Grades 8-10.

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