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

Explanation:

Step1: Analyze Nessa's Proof (ASA)

Nessa used ASA, so she needed two angles and the included side. Given \(\angle B \cong \angle N\), \(\overline{BC} \cong \overline{NM}\) (included side), and \(\angle C \cong \angle M\) (right angles, so congruent). Also, \(\angle A \cong \angle Q\) by third angle theorem (since two angles are congruent, third must be too).

Step2: Analyze Roberto's Proof (AAS)

Roberto used AAS, so two angles and a non - included side. He has \(\angle B \cong \angle N\), \(\angle C \cong \angle M\), and needs a non - included side. The side \(\overline{AB} \cong \overline{QN}\) (opposite right angles, so hypotenuses) would be part of Roberto's proof (AAS: two angles and a non - included side), but not Nessa's (ASA: two angles and included side). The other statements: \(\angle C \cong \angle M\) (used in ASA), \(\overline{BC} \cong \overline{NM}\) (included side in ASA), \(\angle A \cong \angle Q\) (follows from ASA) are part of Nessa's proof. So the statement \(\overline{AB} \cong \overline{QN}\) because they are both opposite a right angle is in Roberto's proof but not Nessa's.

Answer:

The statement " \(\boldsymbol{\overline{AB} \cong \overline{QN}}\) because they are both opposite a right angle" is the one included in Roberto's proof but not Nessa's.