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

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 .

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 .

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

Answer 1

To show that triangles \( \triangle ABD \cong \triangle CBD \) using the SAS (Side-Angle-Side) Congruence Theorem, you need to establish that two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of the other triangle.

In this case, we have:

• Side \( AB \) is congruent to side \( CB \) (because opposite sides of a rhombus are equal).
• Side \( AD \) is congruent to side \( CD \) (for the same reason).
• Angle \( ABD \) is congruent to angle \( CBD \) (because they are vertical angles).

To correctly use the SAS Congruence Theorem, you would find it helpful to know that \( BD \) is the angle bisector of \( \angle ABC \). This way, you can establish that \( \angle ABD \) is equal to \( \angle CBD \) since these angles are bisected.

Thus, the correct answer is:

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