Question

Question

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

(1 point)
Responses which answer is correct and why a b c or d which one is it
∠E≅∠Q
∠E≅∠Q

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

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

∠DCE≅∠DCQ
∠DCE≅∠DCQ
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Answer 1

To determine which piece of information is necessary to prove the triangles congruent using the Angle-Side-Angle (ASA) criterion, we first need to understand what ASA requires. ASA states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Given the options:

1. ∠E≅∠Q - This provides one pair of congruent angles.

2. DE¯¯¯¯¯¯¯¯ ≅ DQ¯¯¯¯¯¯¯¯ - This provides the information about one pair of corresponding sides.

3. CE¯¯¯¯¯¯¯¯ ≅ CQ¯¯¯¯¯¯¯¯ - This provides information about second pair of sides.

4. ∠DCE≅∠DCQ - This provides another pair of congruent angles.

To apply ASA, we already have at least two angles covered and now we need a side that is included between them. If we already with two angles and a congruent side between these angles (like DE and DQ if they are included), we'd have our ASA.

From the options, if we already know at least one angle (say ∠DCE≅∠DCQ) is congruent, we also know one side congruency (like DE≅DQ), then for the third angle, we would have to establish that there is congruency between below two angles (which both provide necessary extra information).

Assuming that angles DCE and DCQ are not congruent, the only congruent option to prove triangle congruence via ASA would be the angle-combination: ∠E≅∠Q.

Thus, the additional information needed can be concluded to be ∠E≅∠Q.

So the correct answer is a: ∠E≅∠Q.