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

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 ∠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 explain why triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent by the SAS (Side-Angle-Side) Congruence Theorem, we need to demonstrate that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle.

1. Identify the Sides:

   • Side \( AB \) is congruent to side \( CB \) (since opposite sides of a rhombus are congruent).
   • Side \( BD \) is common to both triangles \( \triangle ABD \) and \( \triangle CBD \).

2. Include an Angle:

   • The angle we need to ensure is congruent between the two triangles is \( \angle ABD \) and \( \angle CBD \).

For the SAS congruence to hold, it is essential to know that the angles \( \angle BAD \) and \( \angle BCD \) are equal (which they will be in a rhombus as it has congruent opposite angles).

Now, regarding the provided options, the one that indicates that a line is an angle bisector of angles within either triangle would help show that the two angles are indeed equal.

The most relevant choice for establishing angle congruence between triangles \( \triangle ABD \) and \( \triangle CBD \) using SAS is:

"AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BAD."

Confirming that \( AC \) bisects \( \angle BAD \) would provide the information to state that \( \angle ABD \) is equal to \( \angle CBD \), supporting the SAS congruence criteria.