Question

nessa proved that these triangles are congruent using asa. roberto proved that they are congruent using aas. which statement and reason would be included in robertos proof that was not included in nessas proof? given: ∠b ≅ ∠n; bc ≅ nm; ∠c is right; ∠m is right. prove: △abc ≅ △qnm. ∠a ≅ ∠q because of the third - angle theorem. ab ≅ qn because they are both opposite a right angle. bc ≅ nm because it is given. ∠c ≅ ∠m because right angles are congruent.

Explanation:

Step1: Recall ASA and AAS congruence criteria

ASA (Angle - Side - Angle) requires two angles and the included side to be congruent. AAS (Angle - Angle - Side) requires two angles and a non - included side to be congruent.

Step2: Analyze given information

We are given $\angle B\cong\angle N$, $\overline{BC}\cong\overline{NM}$, $\angle C$ is right and $\angle M$ is right. Nessa used ASA. She likely used the two given angles and the given included side.

Step3: Determine what's needed for AAS

For AAS, we need another pair of angles. Since the sum of angles in a triangle is 180 degrees, by the third - angle theorem, if two pairs of angles in two triangles are congruent, the third pair of angles is also congruent. So, $\angle A\cong\angle Q$ because of the third - angle theorem is a statement that would be in Roberto's AAS proof but not necessarily in Nessa's ASA proof.

Answer:

$\angle A\cong\angle Q$ because of the third angle theorem.