In order to prove the triangles pictured below congruent using ASA, what is the one piece of additional information that is needed?

Question

(1 point)
Responses which answer is correct and why

∠E≅∠Q
∠E≅∠Q

DE¯¯¯¯¯¯¯¯ ≅ DQ¯¯¯¯¯¯¯¯
DE¯¯¯¯¯¯¯¯ ≅ DQ¯¯¯¯¯¯¯¯

CE¯¯¯¯¯¯¯¯ ≅ CQ¯¯¯¯¯¯¯¯
CE¯¯¯¯¯¯¯¯ ≅ CQ¯¯¯¯¯¯¯¯

∠DCE≅∠DCQ
∠DCE≅∠DCQ
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Answer 1

To prove two triangles congruent using the ASA (Angle-Side-Angle) criterion, you need two angles and the side included between them to be congruent in both triangles.

Given the options:

1. ∠E≅∠Q (Angle)
2. DE¯¯¯¯¯¯¯¯ ≅ DQ¯¯¯¯¯¯¯¯ (Side)
3. CE¯¯¯¯¯¯¯¯ ≅ CQ¯¯¯¯¯¯¯¯ (Side)
4. ∠DCE≅∠DCQ (Angle)

You already have one pair of equal angles (let's assume it's either ∠E or ∠Q), but you need the following for ASA:

• One more angle that corresponds to the same position in both triangles.
• The side between these two angles must be equal in both triangles.

Based on this reasoning, the correct additional piece of information needed to prove the triangles congruent using ASA would be:

∠DCE≅∠DCQ

Adding this angle will help establish the congruence of the triangles using the ASA criterion. This is because with ∠E ≅ ∠Q and ∠DCE ≅ ∠DCQ, along with DE ≅ DQ (the included side), you have two angles and the included side for both triangles, thus fulfilling the ASA condition.