If △ABC∼△LMN and △LMN∼△XYZ, is △ABC∼△XYZ? Use the AA similarity criterion to support your answer.
review what it means to be similar. Angles are preserved. Equality is transitive.
To determine if △ABC∼△XYZ, we can use the AA similarity criterion, which states that if two triangles have corresponding angles that are congruent, then the triangles are similar.
Given that △ABC∼△LMN and △LMN∼△XYZ, we know that their corresponding angles are congruent. Specifically, we have:
∠A ≅ ∠L (corresponding angles between △ABC and △LMN)
∠B ≅ ∠M (corresponding angles between △ABC and △LMN)
∠C ≅ ∠N (corresponding angles between △ABC and △LMN)
Also:
∠L ≅ ∠X (corresponding angles between △LMN and △XYZ)
∠M ≅ ∠Y (corresponding angles between △LMN and △XYZ)
∠N ≅ ∠Z (corresponding angles between △LMN and △XYZ)
From the given information, we can conclude that:
∠A ≅ ∠X (as ∠A ≅ ∠L and ∠L ≅ ∠X)
∠B ≅ ∠Y (as ∠B ≅ ∠M and ∠M ≅ ∠Y)
∠C ≅ ∠Z (as ∠C ≅ ∠N and ∠N ≅ ∠Z)
Therefore, △ABC∼△XYZ since all corresponding angles are congruent.