Question

the triangles below are similar because of the
aa similarity postulate.
sss similarity theorem.
sas similarity theorem.
the triangles are not similar.

Explanation:

1. First, analyze the angles of each triangle:

• For triangle \( XYZ \), it is a right triangle (angle \( Z = 90^\circ \)) and angle \( Y = 50^\circ \). So the third angle \( X=180^\circ - 90^\circ - 50^\circ=40^\circ \).
• For triangle \( ABC \), it is a right triangle (angle \( C = 90^\circ \)) and angle \( A = 40^\circ \). So the third angle \( B = 180^\circ - 90^\circ - 40^\circ = 50^\circ \).

1. Now, check the similarity postulate:

• The AA (Angle - Angle) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• In triangle \( XYZ \) and triangle \( ABC \), we have two pairs of congruent angles: \( \angle X=\angle A = 40^\circ \) and \( \angle Z=\angle C=90^\circ \) (or \( \angle Y=\angle B = 50^\circ \) and \( \angle Z=\angle C = 90^\circ \)). So by the AA similarity postulate, the triangles are similar.
• The SSS (Side - Side - Side) similarity theorem requires the ratios of all three corresponding sides to be equal, and we don't have side length information here. The SAS (Side - Angle - Side) similarity theorem requires two sides to be in proportion and the included angle to be equal, and we don't have side length information for that either. And we have already shown the triangles are similar, so the "The triangles are not similar" option is incorrect.

Answer:

A. AA similarity postulate