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} \cong \overline{nm}\\); ∠c is right, ∠m is right prove: \\(\triangle abc \cong \triangle qnm\\) \\(\bigcirc\\) ∠a ≅ ∠q because of the third angle theorem. \\(\bigcirc\\) \\(\overline{ab} \cong \overline{qn}\\) because they are both opposite a right angle. \\(\bigcirc\\) \\(\overline{bc} \cong \overline{nm}\\) because it is given. \\(\bigcirc\\) ∠c ≅ ∠m because right angles are congruent.

Explanation:

• ASA (Angle - Side - Angle) for Nessa: To prove congruence by ASA, we need two angles and the included side. We know $\angle B\cong\angle N$, $\overline{BC}\cong\overline{NM}$, and $\angle C\cong\angle M$ (right angles are congruent). So Nessa can use $\angle B$, $\overline{BC}$, $\angle C$ (or $\angle N$, $\overline{NM}$, $\angle M$) for ASA.
• AAS (Angle - Angle - Side) for Roberto: For AAS, we need two angles and a non - included side. We already know $\angle B\cong\angle N$ and $\angle C\cong\angle M$. To get the third angle, we use the Third Angle Theorem, which 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. Then with $\angle A\cong\angle Q$, $\angle B\cong\angle N$, and $\overline{BC}\cong\overline{NM}$ (the side is non - included between the two angles in AAS), Roberto can prove congruence.
• Eliminating other options:
• $\overline{AB}\cong\overline{QN}$ because they are both opposite a right angle: This is not relevant to the difference between ASA and AAS proofs here. Also, we can't assume this without proper proof, and it's not a part of the AAS proof's unique step compared to ASA.
• $\overline{BC}\cong\overline{NM}$ because it is given: This is a given statement and is used in both ASA and AAS proofs, so it's not unique to Roberto's proof.
• $\angle C\cong\angle M$ because right angles are congruent: This is used in Nessa's ASA proof (since $\angle C$ and $\angle M$ are angles) and is not unique to Roberto's AAS proof.

Answer:

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