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

Answer 1

Answer: ∠A ≅ ∠Q because of the third angle theorem.

Answer 2

∠A ≅ ∠Q because of the third angle theorem.

Explanation:

Given information: ∠B ≅ ∠N; BC ≅ NM; ∠C is right; ∠M is right.

Prove: ΔABC ≅ ΔQNM

Nessa's Proof:

In ΔABC and ΔQNM,

∠B ≅ ∠N (Given)

BC ≅ NM (Given)

∠C≅∠M (Right angles)

In both triangle two corresponding angles and there included sides are congruent. By ASA postulate

ΔABC ≅ ΔQNM

Hence proved.

Roberto's Proof:

In ΔABC and ΔQNM,

∠B ≅ ∠N (Given)

∠C≅∠M (Right angles)

∠A≅∠Q (Third angle Theorem)

BC ≅ NM (Given)

In both triangle two corresponding angles and the non-included sides are congruent. By AAS postulate

ΔABC ≅ ΔQNM

Hence proved.

Therefore, the required statement is ∠A ≅ ∠Q and the reason is third angle theorem.