Final answer:
Triangles can be proved similar via the AA Similarity Postulate, while the Law of Sines and Law of Cosines are used to solve triangle problems. For a hypothesis test comparing pass rates of Math 1A and 1B, the alternate hypothesis would suggest a difference in the pass rates.
Step-by-step explanation:
The theorem or postulate used to prove that triangles are similar is the AA Similarity Theorem or Postulate, denoted as (A) in the question. This theorem states that two triangles are similar if they have two corresponding angles that are congruent. The question also involves finding the length of segment AD, which requires additional information not provided in this excerpt. Relevant mathematical laws like the Law of Sines and the Law of Cosines can be used to solve various triangle problems, including finding missing sides or angles in triangles, whether right-angled or not.
To address the hypothesis testing question in the context of statistics, the correct alternate hypothesis, given the information about pass rates for Math 1A and Math 1B, would likely be B. Ha: PA ≠ PB, which states that there is a difference in the pass rates for the two math courses. However, without knowing the specifics of the hypothesis testing structure or the context of the original question, we cannot provide an exact solution to this part.
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