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

Explanation:

For ASA (Angle - Side - Angle), we need two angles and the included side. For AAS (Angle - Angle - Side), we need two angles and a non - included side.
Nessa used ASA. She might have used \(\angle B\cong\angle N\), \(BC\cong NM\), and \(\angle C\cong\angle M\) (since right angles are congruent).
Roberto used AAS. He needed a non - included side. 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) is part of AAS but not of ASA.
\(\overline{BC}\cong\overline{NM}\) is given and is part of both proofs. \(\angle C\cong\angle M\) (right angles congruent) is also part of both proofs. \(\overline{AB}\cong\overline{QN}\) is not relevant as we are not proving sides are congruent in the given congruence criteria context for this problem.

Answer:

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