Question

Which congruence theorem can be used to prove △BDA ≅ △DBC?

Triangles B D A and D B C share side D B. Angles C B D and A D B are right angles. Sides C D and B A are congruent.

HL
SAS
AAS
SSS

Answer 1

To determine which congruence theorem can be used to prove \( \triangle BDA \cong \triangle DBC \), we need to analyze the given information:

1. **Shared side:** Both triangles share side \( DB \).
2. **Right angles:** Angles \( CBD \) and \( ADB \) are right angles.
3. **Congruent sides:** Sides \( CD \) and \( BA \) are congruent.

Given that both are right triangles (due to the right angles), we can apply the **Hypotenuse-Leg (HL) theorem**. The HL theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

In this case:
- The hypotenuses \( AB \) (in \( \triangle BDA \)) and \( BC \) (in \( \triangle DBC \)) are congruent because both are sides across right angles.
- The sides \( BA \) and \( CD \) are given as congruent.

Thus, the **HL theorem** can be used to prove that \( \triangle BDA \cong \triangle DBC \).

The answer is: **HL**.