Use the image to answer the question.

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

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 .

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

Answer 1

To explain why triangle \( \triangle ABD \cong \triangle CBD \) using the SAS (Side-Angle-Side) Congruence Theorem, we need two pairs of sides that are congruent and the angle between them.

In a rhombus, all sides are congruent. Therefore, sides \( AB \) and \( CB \) are congruent since they are opposite sides of the rhombus. The angle \( \angle ABD \) and \( \angle CBD \) will be key.

Among the options provided, the statement that would be most helpful is:

"BD is the angle bisector of \( \angle ABC \)."

This information implies that \( \angle ABD \) is equal to \( \angle CBD \), fulfilling the requirement of SAS (two sides and the included angle being equal) to conclude that \( \triangle ABD \cong \triangle CBD \).