To identify the incorrect step in Misha's proof by contradiction showing that two distinct lines \( l \) and \( n \) with the same slope \( m \) are parallel, we can outline a general structure of a typical proof by contradiction in this context. Here's a possible logical flow:
- Assumption for contradiction: Assume that the lines \( l \) and \( n \) with the same slope \( m \) are not parallel.
- Consequence of assumption: If lines \( l \) and \( n \) are not parallel, then they must intersect at some point.
- Using the slope information: Because both lines have the same slope \( m \), they will either coincide or will be parallel, contradicting the assumption that they intersect.
- Conclusion: Therefore, our initial assumption must be incorrect, and lines \( l \) and \( n \) must be parallel if they have the same slope.
The potential incorrect step often occurs in how the assumption is framed or how the properties of lines with the same slope are applied. The possible incorrect steps could include:
- Misinterpreting the condition of distinct lines: If \( l \) and \( n \) are truly distinct and intersect, they cannot have the same slope \( m \).
- Failing to recognize that lines with the same slope that are distinct are per definition parallel.
Without the exact text of Misha's proof, the common mistake would likely be in step 3, where the distinction between intersection and parallelism could be misapplied, leading to an incorrect interpretation of the relationship between the lines.
To accurately pinpoint the error, reviewing her steps and reasoning regarding the properties of slopes and intersections is crucial. In a correct proof by contradiction, one would not conclude that intersecting lines can share the same slope unless they are coincident. Thus, the contradiction arises when concluding that distinct lines with the same slope cannot intersect.
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