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 Which is the true correct answer and why

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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GPT-4o mini · Answer

To demonstrate that triangles $ \triangle ABD $ and $ \triangle CBD $ are congruent by using the SAS (Side-Angle-Side) Congruence Theorem, we need to show that two sides and the angle between them are congruent in both triangles. Given the properties of a rhombus, we know: 1. Opposite sides are congruent. 2. Adjacent angles are supplementary. 3. The diagonals bisect each other at right angles. In this case, to use SAS, one useful fact would be knowing that the diagonals bisect the angles at the vertices they connect. Therefore, if we consider triangles $ \triangle ABD $ and $ \triangle CBD $: - **Side $ AD $ is congruent to side $ BC $** (since it is a rhombus). - **Side $ BD $** is common to both triangles, as it’s shared. - We would also need the included angle $ \angle ABD $ to be congruent to $ \angle CBD $. For the angle $ \angle ABD $ to be equal to $ \angle CBD $, a potential statement that would support this could be that one of the diagonals (let's say $ AC $) bisects $ \angle BCD $. Therefore, the correct response that would be helpful in this case is: **"AC is the angle bisector of $ \angle BCD $."** This implies that the angles $ \angle ABD $ and $ \angle CBD $ are congruent, allowing us to use the SAS theorem to state that $ \triangle ABD \cong \triangle CBD $.