Question

in the given figure, which lines are parallel? justify your answer. determine the pairs of parallel lines. choose the correct answer below. a. $overleftrightarrow{ce} parallel overleftrightarrow{ch}$ by the converse of the alternate interior angles theorem b. $overleftrightarrow{ce} parallel overleftrightarrow{ek}$ by the converse of the alternate interior angles theorem c. $overleftrightarrow{hk} parallel overleftrightarrow{ch}$ by the converse of the same-side interior angles postulate d. $overleftrightarrow{ch} parallel overleftrightarrow{ek}$ by the converse of the corresponding angles theorem e. $overleftrightarrow{hk} parallel overleftrightarrow{ek}$ by the converse of the alternate exterior angles theorem f. $overleftrightarrow{ce} parallel overleftrightarrow{hk}$ by the converse of the same-side interior angles theorem

Explanation:

To determine parallel lines, we use angle - related theorems (Alternate Interior Angles, Corresponding Angles, etc.). For option F, when we consider lines \(\overline{CE}\) and \(\overline{HK}\), the same - side interior angles formed by a transversal will satisfy the Converse of the Same - Side Interior Angles Theorem (if same - side interior angles are supplementary, lines are parallel). Let's analyze other options:

• Option A: \(\overline{CE}\) and \(\overline{CH}\) are not parallel as they intersect.
• Option B: \(\overline{CE}\) and \(\overline{EK}\) intersect, so not parallel.
• Option C: \(\overline{HK}\) and \(\overline{CH}\) intersect, not parallel.
• Option D: \(\overline{CH}\) and \(\overline{EK}\) - the reasoning about Corresponding Angles does not hold as they are not set up for that theorem.
• Option E: \(\overline{HK}\) and \(\overline{EK}\) intersect, not parallel.

Only option F has lines \(\overline{CE}\parallel\overline{HK}\) justified by the Converse of the Same - Side Interior Angles Theorem.

Answer:

F. \(\overline{CE}\parallel\overline{HK}\) by the Converse of the Same - Side Interior Angles Theorem