Use the image to answer the question.

Question

An illustration shows two congruent triangles having a common vertex.

If point C is the midpoint of both BD¯¯¯¯¯¯¯¯ and AE¯¯¯¯¯¯¯¯ , what additional information would you need to be able to state that △ABC≅△EDC by the SAS Congruence Theorem?

(1 point)
Responses

∠A≅∠E
angle upper A congruent to angle upper E

AB¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯
Modifying above upper A upper B with bar congruent to Modifying above upper E upper D with bar

∠ACB≅∠ECD
angle upper A upper C upper B congruent to angle upper E upper C upper D

BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯

Answer 1

To apply the SAS (Side-Angle-Side) Congruence Theorem, you need to show that two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle.

Given that point C is the midpoint of both \( BD \) and \( AE \), we already know that:

• \( BC \) is congruent to \( DC \) (since C is the midpoint of BD)
• \( AC \) would be one side, but we need the corresponding side from triangle \( EDC \).

To fully establish the SAS criterion for triangle congruence, you need an additional piece of information that includes an angle that is included between the two sides.

The best response that would support the SAS theorem is:

∠ACB ≅ ∠ECD

This indicates that the angle between sides \( AC \) and \( BC \) in triangle \( ABC \) is congruent to the angle between sides \( EC \) and \( DC \) in triangle \( EDC \), which can confirm the triangles are congruent by SAS.