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:

• Nessa used ASA (Angle - Side - Angle). For ASA, she would use two angles and the included side. She already has \(\angle B\cong\angle N\), \(BC\cong NM\), and since \(\angle C\) and \(\angle M\) are right angles (\(\angle C\cong\angle M\)), she can use ASA.
• Roberto used AAS (Angle - Angle - Side). AAS requires two angles and a non - included side. We already know \(\angle B\cong\angle N\) and \(BC\cong NM\). For AAS, we need another pair of angles. The third - angle theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. So, \(\angle A\cong\angle Q\) (by the third - angle theorem) along with \(\angle B\cong\angle N\) and \(BC\cong NM\) can be used for AAS.
• \(\overline{AB}\cong\overline{QN}\) is not relevant as we are not using a side - side - side or hypotenuse - leg (since we don't know if they are hypotenuses in a right - triangle congruence context here) for the given congruence criteria (ASA and AAS).
• \(BC\cong NM\) is given and is used in both ASA and AAS.
• \(\angle C\cong\angle M\) (right angles are congruent) is used in Nessa's ASA proof.

Answer:

\(\angle A\cong\angle Q\) because of the third angle theorem.