Question

find the truth value of the conditional. explain your reasoning, or show a counterexample.
if two lines are parallel, then they do not intersect.

choose the correct answer below.
a. it is possible for parallel lines to intersect, so a counterexample exists. therefore, the conditional is false.
b. the hypothesis is false since two lines cannot be parallel. therefore, the conditional is true.
c. if the conclusion is true and two lines do not intersect, then they are parallel, and so the hypothesis is also true. therefore, the conditional is true.
d. if the hypothesis is true and two lines are parallel, then they do not intersect, and so the conclusion is also true. therefore, the conditional is true.

Explanation:

To determine the truth value of the conditional "If two lines are parallel, then they do not intersect", we analyze each option:

• Option A: Parallel lines (in a plane) never intersect, so this is incorrect.
• Option B: Lines can be parallel (e.g., in Euclidean geometry), so the hypothesis can be true, making this wrong.
• Option C: The reasoning confuses the converse (if not intersect, then parallel) with the original conditional, which is incorrect.
• Option D: By the definition of parallel lines (in a plane, lines that never meet), if the hypothesis (two lines are parallel) is true, the conclusion (they do not intersect) is also true. So the conditional is true.

Answer:

D. If the hypothesis is true and two lines are parallel, then they do not intersect, and so the conclusion is also true. Therefore, the conditional is true.