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 which answer is correct and why by solving it

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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Answer 1

To determine which option would be helpful to know to explain why triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent by the SAS (Side-Angle-Side) Congruence Theorem, we need to identify the corresponding sides and angle between the triangles.

In a rhombus, the diagonals bisect each other at right angles. Therefore:

1. Side BD is a side common to both triangles \( \triangle ABD \) and \( \triangle CBD \).
2. Sides AB and CB are congruent since opposite sides of a rhombus are equal.
3. The angles \( \angle ABD \) and \( \angle CBD \) are both congruent because diagonals of a rhombus bisect the angles.

Thus, we need to show that the included angle (the angle between the known sides) is congruent in both triangles, which leads us to the SAS theorem.

To explain why \( \triangle ABD \cong \triangle CBD \) by SAS, it would be most helpful to know that:

BD is the angle bisector of \( \angle ABE \) (which is the same as \( \angle ABC \)). Thus, it doesn't directly answer the question since the angle bisector needs to refer specifically to angles created with sides of those triangles.

However, what is helpful is knowing:

• Side \( AD \) is congruent to side \( CD \).
• Angle \( ABD \) is equal to angle \( CBD \).

The correct option to select and explain from your provided choices is:

"AC¯ is the angle bisector of \( \angle BAD\)" since it helps demonstrate that \( \angle ABD \) is equal to \( \angle CBD \).

In summary, knowing that \( BD \) is the angle bisector of \( \angle ABC \) is also important but not exactly, as we could conclude triangle congruence by using \( AC \).

Therefore, you would explain:

Response: AC is the angle bisector of \( \angle BAD \) because it would make sure that we can demonstrate that the angles across the triangles are congruent. Thus satisfying the criteria for SAS.