Triangles A B C and N M Q are shown. Sides B C and N M are congruent. Angles A B C and Q N M are congruent. Angles B C A and N M Q are both right angles.

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: AngleB ≅ AngleN; BC ≅ NM; AngleC is right; AngleM is right
Prove: TriangleABC ≅ TriangleQNM

AngleA ≅ AngleQ because of the third angle theorem.
AB ≅ QN because they are both opposite a right angle.
BC ≅ NM because it is given.
AngleC ≅ AngleM because right angles are congruent.

Answer 1

In Roberto's proof using the AAS (Angle-Angle-Side) criterion, he would include a statement regarding the angles of the triangles that would not be explicitly referenced in Nessa's ASA (Angle-Side-Angle) proof.

Specifically, the statement that would be included in Roberto's proof, which is not in Nessa's, is:

Statement: Angle A ≅ Angle Q.
Reason: By the Third Angle Theorem.

This theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. This is a crucial step in the AAS proof method.

Nessa's proof focuses on the congruence given and the right angles (using ASA), hence may not need to state the congruence of the third angle explicitly. However, it would be fundamental in Roberto's AAS reasoning to establish the congruence of all three angles in the triangles.