Final answer:
By definition, parallel lines means alternate interior angles are congruent when cut by a transversal. If the alternate interior angles are not congruent, it indicates that the lines are not parallel. This is based on the principle that if properties of parallel lines do not hold, the lines must be non-parallel.
Step-by-step explanation:
To prove that if alternate interior angles are not congruent then two lines cut by a transversal are not parallel, we use the definition of parallel lines. Parallel lines are two lines in the same plane that do not intersect. One of the properties of parallel lines is that alternate interior angles are congruent when the lines are cut by a transversal.
If alternate interior angles are not congruent, it means that one angle is larger or smaller than the other. If the lines were parallel, the alternate interior angles would match in measurement according to the parallel lines axiom. Since they do not, we can infer that the lines diverge or converge, meaning they are not parallel. Therefore, the logical conclusion is that the lines must be non-parallel.
Always remember to check if your reasoning makes sense, especially when dealing with geometrical concepts. For example, in an interference pattern, angles cannot be greater than 90°, which could serve as a guide in verifying the plausibility of your result.