Question

To prove that Δlmn Δxyz by the SAS similarity theorem, it also needs to be shown that ∠n ≅ ∠z, ∠n ≅ ∠x, ∠l ≅ ∠z, and ∠l ≅ ∠y. Which of the following statements is true?

1) ∠n ≅ ∠z, ∠n ≅ ∠x, ∠l ≅ ∠z, and ∠l ≅ ∠y
2) ∠n ≅ ∠z and ∠l ≅ ∠y
3) ∠n ≅ ∠z and ∠l ≅ ∠z
4) ∠n ≅ ∠x and ∠l ≅ ∠y

Answer 1

The SAS Similarity Theorem requires two pairs of corresponding sides to be proportional and their included angle to be congruent. To prove ΔLMN ≅ ΔXYZ by SAS, it is only necessary to show that ∠N ≅ ∠Z and ∠L ≅ ∠Y, aligning with option 2 of the given statements.

Step-by-step explanation:

To prove that triangles ΔLMN and ΔXYZ are similar by the Side-Angle-Side (SAS) Similarity Theorem, certain conditions must be met. Specifically, two pairs of corresponding sides must be proportional, and the angle between those sides must be congruent. This means that if we want to prove ΔLMN ≅ ΔXYZ using SAS, we only need to show that two pairs of corresponding sides are proportional and their included angles are congruent, not all angles as listed in the options given to the question.

The correct statement, provided that two pairs of corresponding sides have already been shown to be proportional, would be that ∠N ≅ ∠Z and ∠L ≅ ∠Y (option 2). The other pairs of angles (∠N and ∠X or ∠L and ∠Z) do not need to be congruent to satisfy the SAS Similarity Theorem requirement, and proving this is unnecessary if not supported by the proportional sides evidence.