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)ResponsesBD¯¯¯¯¯¯¯¯ 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 .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 ∠BCD.

Answer 1

To use the SAS (Side-Angle-Side) Congruence Theorem to show that triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent, you need two sides and the included angle to be congruent.

In rhombus ABCD, we know the following:

1. \( AB = BC \) (sides of the rhombus),
2. \( AD = DC \) (sides of the rhombus),
3. \( \angle ABD = \angle CBD \) (angles opposite to the sides).

Given the properties of the rhombus, if you want to determine the congruence using the SAS theorem, it is helpful to know that the diagonal \( BD \) bisects angles \( \angle ABC \) and \( \angle ADC \). Specifically, if \( BD \) is the angle bisector of \( \angle ABC \) or \( \angle BDC \), you can use this to establish that the angles \( \angle ABD \) and \( \angle CBD \) are congruent.

Therefore, knowing that \( BD \) is the angle bisector of \( \angle ABC \) would be helpful. The correct response is:

• \( \overline{BD} \) is the angle bisector of \( \angle ABC \).