The SAS Congruence Theorem Quick Check

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Question
Use the image to answer the question.

A rhombus A B C D. From the top, clockwise, the vertices are A, D, C, and B.

If you wanted to explain why △ABD≅CBD by the SAS Congruence Theorem, which of the following would be helpful to know?

(1 point)
Responses

AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.
Modifying above upper A upper C with bar is the angle bisector of angle upper B upper C upper D .

BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠ABC.
Modifying above upper B upper D with bar is the angle bisector of angle upper A upper B upper C .

AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BAD.
Modifying above upper A upper C with bar is the angle bisector of angle upper B upper A upper D .

BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.
Modifying above upper B upper D with bar is the angle bisector of angle upper B upper C upper D .
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Answer 1

To explain why triangles \( \triangle ABD \cong \triangle CBD \) by the SAS (Side-Angle-Side) Congruence Theorem, we need to establish two sides and the included angle are congruent in both triangles.

In this case, we can note the following:

1. \( AB \) is congruent to \( CB \) because the opposite sides of a rhombus are equal.
2. \( BD \) is common to both triangles, making them congruent.
3. The included angle \( \angle ABD \) is equal to \( \angle CBD \) because the opposite angles of a rhombus are equal.

Based on these considerations, if you wanted to further substantiate the angle congruence, knowing that \( AC \) or \( BD \) is the angle bisector of either \( \angle BAD \) or \( \angle BCD \) would be helpful.

The best response here is:

AC is the angle bisector of \( \angle BCD \) (because this would help establish the angles \( \angle ABD \) and \( \angle CBD \) as congruent, aiding in the SAS proof).

This provides evidence that the angle relationships are established correctly.