Final Answer:
Step 3 employs the AA similarity theorem, establishing the congruence of ∠A and ∠Z in triangles ABC and ZYX.
Step 4 introduces AC/XY = BC/YX, utilizing the SAS similarity theorem to support the overall similarity of ΔABC and ΔZYX, alongside the given ratios and right angles. So, the correct options are:
2) ∠B ≅ ∠Y establishes ΔABC ≅ ΔZYX by the SAS similarity theorem.
4) ZX/AC = 2 proves ΔABC ≅ ΔZYX through the SSS similarity theorem.
Explanation:
In the first step, the assertion "∠B ≅ ∠Y" signifies that the two right triangles, ABC and ZYX, share a common right angle and have an equal corresponding angle. This aligns with the Angle-Angle (AA) criterion for similarity. Applying the SAS (Side-Angle-Side) similarity theorem subsequently concludes that the triangles are indeed similar. This is because the shared right angle and the equal angle provide the necessary angle components for the SAS criterion.
Moving to the second step, the statement "ZX/AC = 2" indicates a proportional relationship between the corresponding sides of the triangles. This condition adheres to the Side-Side-Side (SSS) similarity theorem. The ratio of ZX to AC being 2 establishes a consistent scaling factor between the sides, satisfying the SSS criterion for similarity. Consequently, this confirms that ΔABC is similar to ΔZYX.
In summary, the mathematical justification for the similarity of the triangles relies on both angle and side criteria, employing the AA and SSS theorems. These steps provide a rigorous and systematic approach to demonstrating geometric similarity between the two triangles.