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 roberto’s proof that was not included in nessa’s proof? given: ∠b ≅ ∠n; \overline{bc} ≅ \overline{nm}; ∠c is right; ∠m is right prove: △abc ≅ △qnm ○ ∠a ≅ ∠q because of the third angle theorem. ○ \overline{ab} ≅ \overline{qn} because they are both opposite a right angle. ○ \overline{bc} ≅ \overline{nm} because it is given. ○ ∠c ≅ ∠m because right angles are congruent.

Explanation:

Step1: Analyze ASA and AAS

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

Step2: Check each option

• For $\angle A\cong\angle Q$:
• In Nessa's ASA proof (assuming she used $\angle B\cong\angle N$, $\overline{BC}\cong\overline{NM}$, $\angle C\cong\angle M$), $\angle A\cong\angle Q$ is not directly used. But in AAS (where we can use $\angle B\cong\angle N$, $\angle C\cong\angle M$, $\overline{BC}\cong\overline{NM}$ and then by the third - angle theorem $\angle A\cong\angle Q$ for the AAS (two angles and a non - included side) criterion.
• For $\overline{AB}\cong\overline{QN}$: There is no information given to prove this congruence.
• For $\overline{BC}\cong\overline{NM}$: This is given and is part of both ASA and AAS proofs.
• For $\angle C\cong\angle M$: This is given (right angles) and is part of both ASA and AAS proofs.

Answer:

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