What ADDITIONAL information is needed to prove that △ABC≅△EDC

Question

What ADDITIONAL information is needed to prove that △ABC≅△EDC
 using SAS if point C is the midpoint of both BD¯¯¯¯¯¯¯¯  and AE¯¯¯¯¯¯¯¯

(1 point)
Responses which answer is correct and why

∠ACB≅∠ECD
∠ACB≅∠ECD

BD¯¯¯¯¯¯¯¯≅ DC¯¯¯¯¯¯¯¯
BD¯¯¯¯¯¯¯¯≅ DC¯¯¯¯¯¯¯¯

AB¯¯¯¯¯¯¯¯≅ ED¯¯¯¯¯¯¯¯
AB¯¯¯¯¯¯¯¯≅ ED¯¯¯¯¯¯¯¯

∠BAC≅∠DEC
∠BAC≅∠DEC
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GPT-4o mini · Answer

To prove that triangles $ \triangle ABC \cong \triangle EDC $ using the Side-Angle-Side (SAS) congruence criterion, we need to establish the following: 1. One pair of sides must be congruent. 2. The included angle between those two sides must be congruent. 3. Another pair of sides must be congruent. Given that point $ C $ is the midpoint of $ \overline{BD} $ and $ \overline{AE} $, we know that: - $ \overline{BC} \cong \overline{CD} $ (since $ C $ is the midpoint of $ \overline{BD} $). - $ \overline{AC} \cong \overline{CE} $ (since $ C $ is the midpoint of $ \overline{AE} $). Thus, we already have two pairs of congruent sides: $ \overline{BC} \cong \overline{DC} $ and $ \overline{AC} \cong \overline{ED} $, but we still need the included angle to be congruent. Among the provided options, the correct additional information needed is: **∠ACB ≅ ∠ECD** This choice gives us the included angle between the two pairs of congruent sides $ \overline{AB} $ and $ \overline{ED} $, along with the side $ \overline{AC} $ and $ \overline{EC} $ to prove that $ \triangle ABC \cong \triangle EDC $ by SAS. **Summary:** The correct response is **∠ACB ≅ ∠ECD**, because it provides the necessary angle needed to apply the SAS criterion for triangle congruence.