The SAS Congruence Theorem Quick Check

Question

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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 \( \triangle ABD \cong \triangle CBD \) using the SAS (Side-Angle-Side) Congruence Theorem, you need two sides and the angle between them to be congruent in both triangles.

In the context of rhombus \( ABCD \):

• Both triangles share side \( BD \).
• The sides \( AB \) and \( CB \) are congruent since all sides of a rhombus are equal.
• The angles are equal because \( \triangle ABD \) and \( \triangle CBD \) share angle \( \angle ABD \) (or \( \angle CBD \)).

Given these points, the relevant option would be one that confirms that the angle made by the sides leading up to the shared side (in this case, \( BD \)) is congruent.

The helpful piece of information from the options provided is:

BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.

This shows that the angles on either side of \( BD \) are equal, which is necessary for the SAS theorem in this case.

Thus, the correct response regarding the explanation for \( \triangle ABD \cong \triangle CBD \) using the SAS Congruence Theorem would be:

BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.