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

Question

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 $\mathrm{△}BDA\cong \mathrm{△}DBC$, 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 $\mathrm{△}BDA$) and $BC$ (in $\mathrm{△}DBC$) 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 $\mathrm{△}BDA\cong \mathrm{△}DBC$.

The answer is: HL.