The SAS Congruence Theorem Quick Check

Question

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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)
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 .

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

In the case of the triangles \( \triangle ABD \) and \( \triangle CBD \):

• Both triangles share side \( BD \).
• Side \( AB \) is congruent to side \( CB \) because opposite sides of a rhombus are equal.
• The angles \( \angle ABD \) and \( \angle CBD \) are congruent as they are opposite angles formed by the intersection of the diagonal \( BD \).

For the SAS Congruence Theorem, we specifically need to confirm that the angle between the two sides being compared (which is the shared angle in this case) is equal. Knowing that either angle bisector \( AC \) or \( BD \) is also relevant can help us establish the equality of angles, but confirming that \( BD \) splits \( \angle ABC \) equally as its angle bisector leads to an explicit conclusion.

Based on the options provided, the most relevant statement would be:

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

This information confirms that \( \angle ABD \) and \( \angle CBD \) are equal, supporting the application of the SAS theorem for congruence of the triangles.