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?

Question

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

BC¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯
Modifying above upper B upper C with bar congruent to Modifying above upper D upper C with bar

∠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

GPT-4o mini · Answer

To establish that $\triangle ABC \cong \triangle EDC$ by the SAS (Side-Angle-Side) Congruence Theorem, you need to confirm that two sides of one triangle and the included angle are congruent to two sides of the other triangle and the included angle. Since you know that point $C$ is the midpoint of segments $BD$ and $AE$, you already have the following: 1. $BC \cong DC$ (as segments from $C$ to the midpoints $B$ and $D$ are equal). 2. You can establish $AC$ and $EC$ as the sides of the two triangles. To apply the SAS theorem effectively, you must also show that the included angles are congruent. Thus, you need one more piece of information: Since we need the included angles at point $C$ for both triangles, the necessary information to conclude congruence by SAS is: $\angle ACB \cong \angle EDC$ So the response would be: $\angle ACB \cong \angle EDC$ (angle ACB is congruent to angle EDC).