Math Problem Statement
Solution
The question you provided involves understanding the biconditional statement "Two lines are parallel if and only if they do not intersect" and analyzing different conditional statements for their truth value.
Let's break down the statement first:
- Biconditional meaning: "if and only if" means the statement works both ways:
- If two lines are parallel, they do not intersect.
- If two lines do not intersect, they are parallel.
Now, let's analyze each option:
- A. If two lines are parallel, then they do not intersect. This is true because by definition, parallel lines never intersect.
- B. If two lines are not parallel, then they intersect. This is false because non-parallel lines do not always intersect; for example, skew lines (in 3D space) do not intersect but are not parallel.
- C. If two lines do not intersect, then they are parallel. This is true in 2D geometry (as per the biconditional). However, in 3D, skew lines would also fit this condition, so in 3D, it could be false. But since this is usually assumed in 2D geometry, we'll mark it as true.
- D. If two lines do not intersect, then they are parallel. This is false, as it repeats the previous statement but marks it incorrectly.
Thus, the correct answers should be A and C.
Would you like further clarification or have any additional questions?
Here are 5 related questions to explore further:
- What is the difference between conditional and biconditional statements?
- Can two lines that are not parallel ever not intersect in 3D geometry?
- How do skew lines differ from parallel lines?
- What is the contrapositive of a biconditional statement?
- How do intersecting lines differ from non-intersecting lines in geometry?
Tip: Always verify whether a question assumes 2D or 3D geometry, as this can affect the truth of statements about lines.
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Math Problem Analysis
Mathematical Concepts
Biconditional Statements
Parallel Lines
Intersection of Lines
Logic in Geometry
Formulas
-
Theorems
Parallel Line Theorem
Suitable Grade Level
Grades 9-12
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