Bot GPT 4-o mini To determine which congruence theorem can be used to prove that triangles \( \triangle BAE \) and \( \triangle CAD \) are congruent, let's analyze the given information:
- \( \angle BAE \) is a right angle, which means \( \angle BAE = 90^\circ \).
- Sides \( BA \) and \( AC \) are congruent, so \( BA \cong AC \).
- Sides \( BE \) and \( CD \) are also congruent, so \( BE \cong CD \).
We can summarize the information as follows:
- \( BA \cong AC \) (given)
- \( BE \cong CD \) (given)
- \( \angle BAE \cong \angle CAD = 90^\circ \) (both are right angles)
From this information, we can see that we have:
- One pair of corresponding sides is congruent (\( BA \cong AC \)),
- Another pair of corresponding sides is congruent (\( BE \cong CD \)),
- The angle included between these two sides is a right angle for both triangles.
Thus, we can use the SAS (Side-Angle-Side) congruence theorem:
- We have \( BA \) and \( AC \) as sides,
- The angle at \( A \) is a right angle (common angle),
- And the other pair of sides \( BE \) and \( CD \) is also congruent.
Therefore, the congruence theorem that can be used to prove \( \triangle BAE \cong \triangle CAD \) is SAS.
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