Question

which options have redundant(too much) information for proving similarity?*

Explanation:

• ASA (Angle - Side - Angle): In the context of triangle similarity, if two angles and the included side of one triangle are known, and for triangle similarity, we only need two angles (by the AA - Angle - Angle criterion, since the sum of angles in a triangle is \(180^{\circ}\), if two angles are equal, the third is also equal). So, specifying the side (in addition to two angles) is redundant for proving similarity.
• SAS (Side - Angle - Side): For triangle similarity, the SAS similarity criterion requires that the ratio of two sides and the included angle. But if we know the ratio of two sides and the included angle, it is not redundant as it is a valid and non - redundant criterion for similarity (unlike in the case of ASA where the side is extra when considering similarity based on angles).
• AAS (Angle - Angle - Side): Similar to ASA, since by the AA criterion (two angles), we can establish similarity. Knowing the side (in AAS) is extra information as we don't need the side length for similarity (just for congruence in some cases).
• HL (Hypotenuse - Leg): This is a criterion for right - triangle congruence. But for similarity of right - triangles, we can use the AA criterion (one right angle and one other acute angle). So, specifying the hypotenuse and a leg (HL) is redundant for proving similarity.

Answer:

AAS, HL, ASA