Question - Determining Triangle Similarity with Angle-Angle Criterion
Solution:
The question asks which information would be enough to conclude that triangle ABC is similar to triangle MNP. According to the AA (Angle-Angle) criterion, two triangles are similar if two angles of one triangle are equal to two angles of the other triangle. So we only need two pairs of equal angles to conclude similarity.Let's analyze the given options:1) If ∠A = ∠M and ∠C = ∠P, then by the AA criterion, we have two pairs of equal angles, which is sufficient to conclude that triangle ABC is similar to triangle MNP.2) If MP = BC and ∠C = ∠P, even though we have a side and an angle, without a corresponding angle-angle pair, or side-angle pair where the sides are proportional, we cannot be sure of the similarity.3) If ∠A = ∠M and AB/MN = AC/MP, this option provides one pair of angles and a proportion between sides. However, the given proportion does not establish a corresponding angle-angle pair nor does it satisfy the SAS (Side-Angle-Side) similarity criterion as the given angle is not included between the sides of the proportion.4) If ∠B = ∠N and AB = MN, this provides one pair of angles and one pair of sides equal in length. However, just like the previous option, it does not satisfy the criteria for similarity on its own (either AA, SAS where the angle must be between the two sides, or SSS with proportional sides).Therefore, the correct answer is:1) If ∠A = ∠M and ∠C = ∠P, as this satisfies the AA criterion for triangle similarity.
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