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; \overline{bc} ≅ \overline{nm}; ∠c is right; ∠m is right prove: △abc ≅ △qnm \bigcirc ∠a ≅ ∠q because of the third angle theorem. \bigcirc \overline{ab} ≅ \overline{qn} because they are both opposite a right angle. \bigcirc \overline{bc} ≅ \overline{nm} because it is given. \bigcirc ∠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 the given information

We are given $\angle B\cong\angle N$, $\overline{BC}\cong\overline{NM}$, $\angle C$ (right angle) and $\angle M$ (right angle, so $\angle C\cong\angle M$).
For ASA (Nessa's proof), we can use $\angle B\cong\angle N$, $\overline{BC}\cong\overline{NM}$, $\angle C\cong\angle M$.
For AAS (Roberto's proof), we need another pair of angles. Since the sum of angles in a triangle is $180^{\circ}$, if two angles of one triangle are congruent to two angles of another triangle, the third angles are congruent (third angle theorem). So $\angle A\cong\angle Q$ (using the third angle theorem) is part of AAS but not part of ASA.

Step3: Check other options

• $\overline{AB}\cong\overline{QN}$ is not a given or derivable from basic congruence rules in the context of proving congruence here.
• $\overline{BC}\cong\overline{NM}$ is given and is part of both ASA and AAS.
• $\angle C\cong\angle M$ (right angles are congruent) is part of both ASA and AAS.

Answer:

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