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The question asks whether triangles \( \triangle ABD \) and \( \triangle BCD \) are congruent, based on the diagram shown.

Let's analyze the information:

1. The quadrilateral is a rectangle (as suggested by the right angles and equal opposite sides). This makes sides \( AB = CD = 8 \) and \( AD = BC = 6 \).
2. The line \( BD \) acts as a diagonal dividing the rectangle into two right triangles, \( \triangle ABD \) and \( \triangle BCD \).

### Step-by-Step Analysis for Congruence:
- **Side \( AB = CD = 8 \)**: Both triangles share a corresponding side.
- **Side \( AD = BC = 6 \)**: Both triangles have another corresponding side.
- **Shared side \( BD \)**: Both triangles share the hypotenuse, making it equal.

Given that all three sides of \( \triangle ABD \) and \( \triangle BCD \) are equal, by the **SSS (Side-Side-Side) Congruence Criterion**, the two triangles are congruent.

### Conclusion:
Yes, \( \triangle ABD \) and \( \triangle BCD \) are congruent because they satisfy the SSS congruence criterion.

Do you want more details or have any questions?

### Related Questions:
1. What are the properties of a rectangle that help us conclude the triangles are congruent?
2. Can two triangles be congruent if only two sides are equal but not the angles?
3. How does the Pythagorean theorem apply to right triangles like \( \triangle ABD \) and \( \triangle BCD \)?
4. What is the significance of the diagonal \( BD \) in the congruence of these triangles?
5. Can you identify another geometric shape where triangles formed by a diagonal would also be congruent?

**Tip**: To determine if triangles are congruent, always look for corresponding sides and angles that match, or use congruence postulates like SSS, SAS, or ASA.

Are the triangles ABD and BCD on the right congruent? Give a reason for your answer.

Explore how triangles ABD and BCD are congruent using the Side-Side-Side (SSS) theorem. Analyze the properties of the rectangle and the right triangles formed by its diagonal. Ideal for students in Grades 8-10.

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