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

Explanation:

• ASA (Angle - Side - Angle) requires two angles and the included side. In the given problem, for ASA, we can use \(\angle B\cong\angle N\), \(\overline{BC}\cong\overline{NM}\), and \(\angle C\cong\angle M\) (right angles).
• AAS (Angle - Angle - Side) requires two angles and a non - included side. We already know \(\angle B\cong\angle N\) and \(\angle C\cong\angle M\). By the third - angle theorem (the sum of angles in a triangle is \(180^{\circ}\), if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent), \(\angle A\cong\angle Q\). Then we can use \(\angle B\cong\angle N\), \(\angle A\cong\angle Q\), and \(\overline{BC}\cong\overline{NM}\) for AAS. The statement \(\angle A\cong\angle Q\) because of the third - angle theorem is part of AAS proof but not of ASA proof (in ASA, we don't need to use the third - angle theorem as we are using the two given angles and the included side directly).

Answer:

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